diff --git a/lib/autsr.gi b/lib/autsr.gi
index 29f1181e83..de67d2d35c 100644
--- a/lib/autsr.gi
+++ b/lib/autsr.gi
@@ -620,8 +620,7 @@ local ff,r,d,ser,u,v,i,j,k,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
 
     # do we use induced radical automorphisms to help next step?
     if Size(KernelOfMultiplicativeGeneralMapping(hom))>1 and
-      Size(A)>10^8 and (IsAbelian(r) or
-      Length(AbelianInvariants(r))<10)
+      Size(A)>10^8 and (IsAbelian(r) or AbelianRank(r)<10)
       #(
       ## potentially large GL
       #Size(GL(Length(MPcgs),RelativeOrders(MPcgs)[1]))>10^10 and
diff --git a/lib/grp.gd b/lib/grp.gd
index 26269be9c1..264d4e866d 100644
--- a/lib/grp.gd
+++ b/lib/grp.gd
@@ -861,6 +861,10 @@ InstallTrueMethod( IsPolycyclicGroup,
 ##
 DeclareAttribute( "AbelianInvariants", IsGroup );
 
+# minimal number of generators for abelianization. Used internally to check
+# whether it is worth to attempt to reduce generator number
+DeclareAttribute( "AbelianRank", IsGroup );
+
 #############################################################################
 ##
 #A  IsInfiniteAbelianizationGroup( <G> )
diff --git a/lib/grp.gi b/lib/grp.gi
index d4aa4ae6ce..6b2ca3bae8 100644
--- a/lib/grp.gi
+++ b/lib/grp.gi
@@ -701,6 +701,18 @@ InstallMethod( AbelianInvariants,
     return inv;
     end );
 
+InstallMethod( AbelianRank ,"generic method for groups", [ IsGroup ],0,
+function(G)
+local a,r;
+  a:=AbelianInvariants(G);
+  r:=Number(a,IsZero);
+  a:=Filtered(a,x->not IsZero(x));
+  if Length(a)=0 then return r; fi;
+  a:=List(Set(a,SmallestRootInt),p->Number(a,x->x mod p=0));
+  return r+Maximum(a);
+end);
+
+
 #############################################################################
 ##
 #M  IsInfiniteAbelianizationGroup( <G> ) 
diff --git a/lib/grppclat.gi b/lib/grppclat.gi
index 159a362714..5b9380e46a 100644
--- a/lib/grppclat.gi
+++ b/lib/grppclat.gi
@@ -1201,7 +1201,7 @@ InstallMethod(CharacteristicSubgroups,"solvable, automorphisms",true,
   [IsGroup and IsSolvableGroup],0,
 function(G)
 local A,s;
-  if Length(AbelianInvariants(G))<5 then
+  if AbelianRank(G)<5 then
     TryNextMethod();
   fi;
   A:=AutomorphismGroup(G);
diff --git a/lib/morpheus.gi b/lib/morpheus.gi
index 8c53229076..da48a8dad7 100644
--- a/lib/morpheus.gi
+++ b/lib/morpheus.gi
@@ -1267,8 +1267,7 @@ local G,cl,lcl,len,comb,combc,com,a,cnt,s,alltwo;
   #create just a list of ordinary classes.
   lcl:=List(cl,i->Concatenation(List(i,j->j.classes)));
   len:=1;
-  len:=Maximum(1,Length(MinimalGeneratingSet(
-		    Image(IsomorphismPcGroup((G/DerivedSubgroup(G))))))-1);
+  len:=Maximum(1,AbelianRank(G)-1);
   while true do
     len:=len+1;
     Info(InfoMorph,2,"Trying length ",len);
@@ -2188,8 +2187,8 @@ local m;
     return fail;
   fi;
 
-  if (Length(AbelianInvariants(G))>2 or Length(SmallGeneratingSet(G))>2) and Size(RadicalGroup(G))>1 then
-    # In place until a proper implementation of Cannon/Holt automorphism is
+  if (AbelianRank(G)>2 or Length(SmallGeneratingSet(G))>2) and Size(RadicalGroup(G))>1 then
+    # In place until a proper implementation of Cannon/Holt isomorphism is
     # made available.
     return PatheticIsomorphism(G,H);
   fi;