diff --git a/docs/src/Groups/group_characters.md b/docs/src/Groups/group_characters.md
index 1f7f0349ef55..9e94e965fc11 100644
--- a/docs/src/Groups/group_characters.md
+++ b/docs/src/Groups/group_characters.md
@@ -206,6 +206,7 @@ class_multiplication_coefficient
 known_class_fusion
 order(tbl::GAPGroupCharacterTable)
 possible_class_fusions
+approximate_class_fusion
 ```
 
 ## Character tables and normal subgroups
diff --git a/docs/src/NumberTheory/abelian_closure.md b/docs/src/NumberTheory/abelian_closure.md
index 93b135d8792c..4b0ce54c8c4f 100644
--- a/docs/src/NumberTheory/abelian_closure.md
+++ b/docs/src/NumberTheory/abelian_closure.md
@@ -21,6 +21,8 @@ Given the abelian closure, the generator can be recovered as follows:
 
 ```@docs
 gen(::QQAbField)
+atlas_irrationality
+atlas_description
 ```
 
 ## Printing
diff --git a/src/GAP/wrappers.jl b/src/GAP/wrappers.jl
index f0b05e1b8ffe..cb618551c3b7 100644
--- a/src/GAP/wrappers.jl
+++ b/src/GAP/wrappers.jl
@@ -92,6 +92,7 @@ GAP.@wrap Image(x::Any, y::Any)::GapObj
 GAP.@wrap Image(x::Any)::GapObj
 GAP.@wrap ImmutableMatrix(x::GapObj, y::GapObj, z::Bool)::GapObj
 GAP.@wrap IndependentGeneratorExponents(x::Any, y::Any)::GapObj
+GAP.@wrap InitFusion(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap Indeterminate(x::GapObj)::GapObj
 GAP.@wrap Indeterminate(x::GapObj, y::GAP.Obj)::GapObj
 GAP.@wrap IndeterminateNumberOfUnivariateRationalFunction(x::GapObj)::Int
diff --git a/src/Groups/group_characters.jl b/src/Groups/group_characters.jl
index 2f3f77695c62..d3d323268b89 100644
--- a/src/Groups/group_characters.jl
+++ b/src/Groups/group_characters.jl
@@ -19,7 +19,7 @@
 ##
 ##  Atlas irrationalities
 ##
-"""
+@doc raw"""
     atlas_irrationality([F::AnticNumberField, ]description::String)
 
 Return the value encoded by `description`.
@@ -30,6 +30,7 @@ if `F` is not given then the result has type `QQAbElem`.
 `description` is assumed to have the format defined in
 [CCNPW85](@cite), Chapter 6, Section 10.
 
+# Examples
 ```jldoctest
 julia> Oscar.with_unicode() do
          show(atlas_irrationality("r5"))
@@ -65,6 +66,34 @@ function atlas_irrationality(description::String)
 end
 
 
+@doc raw"""
+    atlas_description(val::QQAbElem)
+
+Return a string in the format defined in
+[CCNPW85](@cite), Chapter 6, Section 10,
+describing `val`.
+Applying [`atlas_irrationality`](@ref) to the result yields `val`.
+
+# Examples
+```jldoctest
+julia> K, z = abelian_closure(QQ);
+
+julia> val = z(5) + z(5)^4;
+
+julia> str = Oscar.atlas_description(val)
+"b5"
+
+julia> val == atlas_irrationality(str)
+true
+```
+"""
+function atlas_description(val::QQAbElem)
+    iso = Oscar.iso_oscar_gap(parent(val))
+    val = iso(val)
+    return string(GAP.Globals.CTblLib.StringOfAtlasIrrationality(val))
+end
+
+
 #############################################################################
 ##
 ##  character tables
@@ -1264,10 +1293,50 @@ end
 
 class_multiplication_coefficient(tbl::GAPGroupCharacterTable, i::Int, j::Int, k::Int) = class_multiplication_coefficient(ZZRingElem, tbl, i, j, k)
 
+@doc raw"""
+    approximate_class_fusion(subtbl::GAPGroupCharacterTable,
+                             tbl::GAPGroupCharacterTable)
+
+Compute for each class of `subtbl` all those classes in `tbl` to which it can
+fuse under an embedding of the group of `subtbl` into the group of `tbl`,
+according to element orders and centralizer orders in the two tables.
+
+If no embedding is possible then return an empty vector.
+Otherwise return a vector of length equal to the number of classes of
+`subtbl`, such that the entry at position $i$ either an integer (if there is
+a unique possible image class) or the vector of the positions of possible
+image classes.
+
+# Examples
+```jldoctest
+julia> subtbl = character_table("A5"); tbl = character_table("A6");
+
+julia> println(approximate_class_fusion(subtbl, tbl))
+Union{Int64, Vector{Int64}}[1, 2, [3, 4], [6, 7], [6, 7]]
+
+```
+"""
+function approximate_class_fusion(subtbl::GAPGroupCharacterTable,
+                                  tbl::GAPGroupCharacterTable)
+  fus = GAPWrap.InitFusion(GAPTable(subtbl), GAPTable(tbl))
+  res = Union{Int, Vector{Int}}[]
+  fus == GAP.Globals.fail && return res
+  for i in 1:length(fus)
+    if fus[i] isa Int
+      push!(res, fus[i])
+    else
+      push!(res, Vector{Int}(fus[i]))
+    end
+  end
+  return res
+end
+
+
 @doc raw"""
     possible_class_fusions(subtbl::GAPGroupCharacterTable,
                            tbl::GAPGroupCharacterTable;
-                           decompose::Bool = true)
+                           decompose::Bool = true,
+                           fusionmap::Vector = [])
 
 Return the array of possible class fusions from `subtbl` to `tbl`.
 Each entry is an array of positive integers, where the value at position `i`
@@ -1281,6 +1350,9 @@ is not checked;
 this does not change the result,
 but in certain situations it is faster to omit this step.
 
+If `fusionmap` is set to a vector of integers and integer vectors then
+only those maps are returned that are compatible with the prescribed value.
+
 # Examples
 ```jldoctest
 julia> possible_class_fusions(character_table("A5"), character_table("A6"))
@@ -1293,9 +1365,14 @@ julia> possible_class_fusions(character_table("A5"), character_table("A6"))
 """
 function possible_class_fusions(subtbl::GAPGroupCharacterTable,
                                 tbl::GAPGroupCharacterTable;
-                                decompose::Bool = true)
+                                decompose::Bool = true,
+                                fusionmap::Vector = [])
+  cond = Dict{Symbol, Any}(:decompose => decompose)
+  if length(fusionmap) != 0
+    cond[:fusionmap] = GapObj(fusionmap, recursive = true)
+  end
   fus = GAPWrap.PossibleClassFusions(GAPTable(subtbl), GAPTable(tbl),
-            GapObj(Dict(:decompose => decompose)))
+            GapObj(cond))
   return [Vector{Int}(x::GapObj) for x in fus]
 end
 
diff --git a/src/Groups/libraries/libraries.jl b/src/Groups/libraries/libraries.jl
index ae66201c4f64..98be4046d2ca 100644
--- a/src/Groups/libraries/libraries.jl
+++ b/src/Groups/libraries/libraries.jl
@@ -26,15 +26,15 @@ end
 function __init_group_libraries()
   props = [
     is_abelian => GAP.Globals.IsAbelian,
-    is_almostsimple => GAP.Globals.IsAlmostSimpleGroup,
+    is_almostsimple => GAP.Globals.IsAlmostSimple,
     is_cyclic => GAP.Globals.IsCyclic,
-    is_nilpotent => GAP.Globals.IsNilpotentGroup,
-    is_perfect => GAP.Globals.IsPerfectGroup,
+    is_nilpotent => GAP.Globals.IsNilpotent,
+    is_perfect => GAP.Globals.IsPerfect,
     is_quasisimple => GAP.Globals.IsQuasisimple,
-    is_simple => GAP.Globals.IsSimpleGroup,
+    is_simple => GAP.Globals.IsSimple,
     is_sporadic_simple => GAP.Globals.IsSporadicSimple,
-    is_solvable => GAP.Globals.IsSolvableGroup,
-    is_supersolvable => GAP.Globals.IsSupersolvableGroup,
+    is_solvable => GAP.Globals.IsSolvable,
+    is_supersolvable => GAP.Globals.IsSupersolvable,
   ]
 
   empty!(_group_filter_attrs)
diff --git a/src/exports.jl b/src/exports.jl
index 5f0af46c1b87..129bb9e6a22a 100644
--- a/src/exports.jl
+++ b/src/exports.jl
@@ -241,12 +241,14 @@ export anti_symmetric_parts
 export anticanonical_bundle
 export anticanonical_divisor
 export anticanonical_divisor_class
+export approximate_class_fusion
 export archimedean_solid
 export as_dictionary
 export as_gset
 export as_perm_group
 export as_polycyclic_group
 export associahedron
+export atlas_description
 export atlas_group
 export atlas_irrationality
 export atlas_program
diff --git a/test/Groups/group_characters.jl b/test/Groups/group_characters.jl
index 9a63e2e62c2c..4973ed130851 100644
--- a/test/Groups/group_characters.jl
+++ b/test/Groups/group_characters.jl
@@ -956,6 +956,10 @@ end
   fus1 = possible_class_fusions(subtbl, tbl)
   fus2 = possible_class_fusions(subtbl, tbl, decompose = false)
   @test fus1 == fus2
+  fus3 = possible_class_fusions(subtbl, tbl, fusionmap = fus1[1])
+  @test length(fus3) == 1 && fus3[1] == fus1[1]
+  @test approximate_class_fusion(subtbl, tbl) == [1, 2, [3, 4], [6, 7], [6, 7]]
+  @test approximate_class_fusion(tbl, subtbl) == []
 end
 
 @testset "normal subgroups" begin
diff --git a/test/Groups/libraries.jl b/test/Groups/libraries.jl
index 804c8bd42f3d..a8040184b337 100644
--- a/test/Groups/libraries.jl
+++ b/test/Groups/libraries.jl
@@ -29,7 +29,9 @@
       is_cyclic,
       is_nilpotent,
       is_perfect,
+      is_quasisimple,
       is_simple,
+      is_sporadic_simple,
       is_solvable,
       is_supersolvable,
       is_transitive,
diff --git a/test/Rings/AbelianClosure.jl b/test/Rings/AbelianClosure.jl
index 8fce53d72386..d40d5ec3a562 100644
--- a/test/Rings/AbelianClosure.jl
+++ b/test/Rings/AbelianClosure.jl
@@ -366,4 +366,15 @@ end
     @test F isa AnticNumberField
     @test dim(F) == 8
   end
+
+  @testset "Atlas irrationalities" begin
+    K, z = abelian_closure(QQ)
+    vals = [z(7) + z(7)^2 + z(7)^4,
+            z(8) + z(8)^3,
+            z(9) + z(9)^-1,
+            z(3) - z(3)^2]
+    for val in vals
+      @test atlas_irrationality(atlas_description(val)) == val
+    end
+  end
 end