need a 3-group with 3 generators

Question

I need a $3$-group of nilpotency class $3$ with $3$ generators, but I can't find any group meeting these criteria. Does such a group exist?

I've already used GAP-System as far as my PC can handle, and I came up with nothing. I tried a small group of order $3^n,n\in\{1,2,3,4,5,6,7,8,9,10\}$. I tried this code in GAP:

for n in [1..10] do
  for j in [1..NrSmallGroups(3^n)] do
    G:=SmallGroup(3^n,j);
    if NilpotencyClassOfGroup(G)=3 then
      if Length(GeneratorsOfGroup(G))=3 then
        Print(n,", ",j,"\n");
      fi;
    fi;
  od;
od;

Answer 1

I think the issue is your formulation and test of "with three generators". Presumably you mean that the group can be generated by 3 elements (but not by 2). Of course you could generate such a group with more elements. The generating set stored by GAP and returned by GeneratorsOfGroup is for solvable groups a polycyclic generating set, which for a group of order $p^n$ has always $n$ generators. Thus your test eliminates all groups.

If you replace Length(GeneratorsOfGroup(G)) by Length(MinimalGeneratingSet(G)) you will find the groups you're interested in (starting in order $3^5$ as noted by Alexander Konovalov above.)