[GAP Forum] 2 -groups of order 2^10

[Bettina Eick]

Dear GAP Forum, dear Reza Orfi,

> I need the complete list of 2-groups of order 2^10 with
> trivial Schur multiplier generated by three elements (not fewer).

You could try to determine such a list with GAP.

First, you could try to compute all 3-generator groups of order 2^10.
For this purpose you have to loop over all 3-generator groups of order
dividing 2^9 (these are available in the SmallGroups library) and
compute their descendants of order 2^10 (the ANUPQ package contains
a method for this purpose).

Then, you could loop over your new list of 3-generator groups of order
2^10 and check which of them have trivial Schur multiplicator. (There
is a function 'AbelianInvariantsSchurMultiplier' available for this
purpose in GAP.)

However, there is a caveat: both parts of this computation will
probably be time- and space-consuming computations. So you may
need serious resources to perform them. Perhaps you could cut down
the number of 3-generator groups of order dividing 2^9 that you
have to consider using theoretical arguments?

Best wishes,
            Bettina