Abstract
Let G be a finite group. The real genus p(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. We show that if G is a group of order 2m that has real genus congruent to 5 (mod 8), then either G has exponent 2m~3 and a dihedral subgroup of index 4 or else the exponent of G is 2m~2. It follows that there are at most 53 isomorphism types of 2-groups with real genus congruent to 5 (mod 8). A consequence of this bound is that almost all positive integers that are the real genus of a 2-group are congruent to 1 (mod 8). We also prove that almost all positive integers that are the real genus of a 2-group with Frattini-class 2 are in a set that has density zero in the positive integers.
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