[GAP Forum] Creating groups by a multiplication rule

[Могильных Иван Юрьевич]

Dear Gap forum,

I have a problem and I wonder if I can handle it with the help of GAP.
Consider a collection G of elements of Z_2^15 (not a subgroup) of size 2048.
Define an operation in the following manner: x*y=x+y^pi_x, where + is the
ordinary addition in Z_2.
Here pi_x is the permutation on {1..n} attached to the element x by some
rule.
Permutations can vary from one element to another.
I have certain values for these permutation, such that (G,*) is a group.
I want to realize this group in computer and carry out manipulations with
it (for example, find subgroups of index 2).
Can this be done using GAP?
Any such group can be generated by a certain set of 35 elements.

So, generally speaking, the problem I faced was creating a group by some
multiplication rule.
According to the GAP documentation, it can be done using domains. However,
I could not succeed in creating new domain with operation.
I could not find an appropriate example, neither.
I would be grateful, if someone helps me with any of the questions above.

Best wishes,
Ivan Mogilnykh