[GAP Forum] PQuotients - meaning???

[Michael Hartley]

I am trying to find all groups of order 2^8 which are quotients of a 
certain infinite FP group.
I know I could use GQuotients combined with SmallGroup, but checking 
*every* small group
of order 256 takes a long time. And 2^8 is actually a testbed for the 
real problems - 2^9 and 2^10.

I read up on Quotient Methods, and got a feeling the PQuotient may be 
the method I need.

I tried the following -

gap> F := FreeGroup("a","b","c");
<free group on the generators [ a, b, c ]>
gap> a := F.1;; b := F.2;; c := F.3;;
gap> rels := [a^2, b^2, c^2, (a*c)^2];;
gap> W := F/rels;
<fp group on the generators [ a, b, c ]>
gap> PQuotient(W,2,8);
<2-quotient system of 2-class 8 with 55 generators>
gap> EpimorphismQuotientSystem(last);
[ a, b, c ] -> [ a1, a2, a3 ]
gap> Size(Image(last));
36028797018963968
gap> 2^55;
36028797018963968

Can someone help me understand what all this means? Why is my quotient 
of order 2^55, and what does that have to do with the parameter '8' I 
passed to PQuotient? Is this order 2^55 group a covering group for *all* 
the Quotients of the form I want  (this would be nice), or is it 
something else? Finally, can PQuotient be wrangled to give what I want - 
a list of all the quotients of order 256 of my FP group W?

Yours, Mike H...