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
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