[GAP Forum] About 3-group maximal class of of order 243 of a strange type

[Benjamin Sambale]

Dear Siddhartha,

I once wrote the following code:

Gamma1:=function(G)
     local LC,f;
     LC:=LowerCentralSeries(G);
     f:=NaturalHomomorphismByNormalSubgroupNC(G,LC[4]);
     return PreImage(f,Centralizer(G/LC[4],Image(f,LC[2])));
end;

Please check if this is your G_1.
Assuming you want groups of maximal class of order *243* with only two 
conjugacy classes of order 9 elements outside G_1, then there is only 
one such group: [243,28].

Best wishes,
Benjamin

Am 22.08.19 um 01:10 schrieb Siddhartha Sarkar:
> Dear forum,
>
> A 3-group G of maximal class of order 343 has a unique two step centraliser
> G_1. There are six conjugacy classes in $G \setminus G_1$ ($p(p-1)$ for
> arbitrary $p$ while degree of commutativity is larger than 1).
>
> I am trying to check if there are such groups with only two conjugacy
> classes that consists of order 9 elements.
>
> In general, are there methods to code the two step centraliser without
> going into group actions? If so, I can only think of brute force checking
> conjugacy over the 162 uniform elements. Is there a better way of doing it?
>
> The same questions applicable for larger prime, groups of order larger than
> $p^{p+2}$ whether there exists only $(p-1)$ many conjugacy classes with
> order $p^2$ elements? I checked that I have to work with groups G of
> maximal class G with nil potency class of G_1 above 3, which make the Hall
> collection formula unpleasant.
>
> Thanks in advance,
> Siddhartha
> _______________________________________________
> Forum mailing list
> Forum at gap-system.org
> https://mail.gap-system.org/mailman/listinfo/forum
>