More convenient GAP code to verify Additional property of d-maximal groups

Question

Let $G$ be a finite $p$-group and $d(G)$ be its minimal number of generators. We say that $G$ is $d$-maximal if $d(H) < d(G)$ for all $H < G$.

The following code determines weather $G$ is $d$-maximal or not.

IsDMaximal := function( G )
local nr;
nr := Length(Filtered(ConjugacyClassesSubgroups(G), x-> Length(MinimalGeneratingSet(Representative(x))) < Length(MinimalGeneratingSet(G))));
return nr = (Length(ConjugacyClassesSubgroups(G)) - 1);
end;

I realized that if $d(H) < d(G)$ for all subgroups $H$ containing the commutator subgroup of $G$, then then $G$ is $d$-maximal. The latter property allows us, in principle, to decrease the execution time for checking the $d$-maximality of $G$ (by testing less number of subgroups), I coded this "property" as follows:

IsDMaximalByComm := function(G)
local ContComms, H, HC;
 ContComms := [];
for H in AllSubgroups(G) do 
      if IsSubgroup(H, DerivedSubgroup(G)) then
         Add(ContComms, H);
         fi;       
od;
for HC in ContComms do 
  if Length(MinimalGeneratingSet(HC)) < Length(MinimalGeneratingSet(G)) then
return G;
fi;
od;
end;

Unfortunately, the execution time got longer!

How to improve my code?

Thanks in advance!

Answer 1

First (as already mentioned in the comments) you could replace

for H in AllSubgroups(G) do 
      if IsSubgroup(H, DerivedSubgroup(G)) then
         Add(ContComms, H);
         fi;       
od;

by

nat:=NaturalHomomorphismByNormalSubgroup(G,DerivedSubgroup(G));
ContComms:=AllSubgroups(Image(nat,G));
ContComms:=List(ContComms,x->PreImage(nat,x));

to only consider subgroups containing the derived subgroup.

The length of a minimal generating set of $G$ is fixed, so store

lm:=Length(MinimalGeneratingSet(G));

Then also you don't need an explicit minimal generating set, but only the length. Thus replace the test

Length(MinimalGeneratingSet(HC)) < Length(MinimalGeneratingSet(G)) then

by

LogInt(IndexNC(HC,DerivedSubgroup(HC)),p)

Finally your logic is wrong -- you return the group as soon as you have found one subgroup with a smaller generating set, but you want to test whether all have (or one is an opposite. Thus

for HC in ContComms do 
  if LogInt(IndexNC(HC,DerivedSubgroup(HC)),p)>lm then return false;fi;
od;
return true;