The real genus of 2-groups

Let G be a finite group. The real genus.( G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we consider 2-groups acting on bordered Klein surfaces. The main focus is determining the real genus of each of the 51 groups of order 32. We also obtain some general results about the partial presentations that 2-groups acting on bordered surfaces must have. In addition, we obtain genus formulas for some families of 2-groups and show that if G is a 2-group with positive real genus, then rho(G) = 1 mod 4.