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

[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