[GAP Forum] coclass 2-groups

[Alexander Konovalov]

Dear Siddhartha,

On 31 Mar 2009, at 12:02, Siddhartha Sarkar wrote:

>  Dear GAP forum,
>>
>> I am trying to access the group of order 2^8 which has the  
>> descripton :
>>
>> (1) it is of co-class 2,
>> (2) cited as G_20 and from family 8 (in the paper by Baginski,  
>> Konovalov
>> "On 2-groups of Almost Maximal Class"); maybe these notations are  
>> standard.

No, we used the numeration for families following the paper by M. F.  
Newman and
E. A. O’Brien, Classifying 2-groups by coclass. Trans. A.M.S., vol. 
351, No.1,
January 1999, 131-169, but the numbering of groups is only used within  
our paper.

>> How to find this in the small group library?
>>
> The presentation is :
>
> Generators : x_1, x_2, y
>
> Relations : x_1^8 = x_2^8 = 1, y^4 = x_2^4, x_1^y = x_1 * x_2, x_2^y =
> x_1^{-2} * x_2^3, [x_2, x_1] = 1

This is the group [ 256, 519 ]:

gap> f:=FreeGroup("x_1","x_2","y");
<free group on the generators [ x_1, x_2, y ]>
gap> AssignGeneratorVariables(f);
#I  Assigned the global variables [ x_1, x_2, y ]
gap> r:=[x_1^8,x_2^8,y^4*x_2^4,y^-1*x_1*y*x_2^-1*x_1^-1,
 > y^-1*x_2*y*x_2^-3*x_1^2,x_1^-1*x_2^-1*x_1*x_2];
[ x_1^8, x_2^8, y^4*x_2^4, y^-1*x_1*y*x_2^-1*x_1^-1,  
y^-1*x_2*y*x_2^-3*x_1^2,
   x_1^-1*x_2^-1*x_1*x_2 ]
gap> G:=f/r;
<fp group on the generators [ x_1, x_2, y ]>
gap> IdGroup(G);
[ 256, 519 ]

Please let me know if you will have further questions.

Best wishes,
Alexander