Finite symmetries of surfaces of p-groups of co-class 1

The genus spectrum of a finite group G is a set of integers g >= 2 such that G acts on a closed orientable compact surface Sigma(g) of genus g preserving the orientation. In this paper, we complete the full classification of spectrum sets of finite p-groups of co-class 1, where p is an odd prime. As a consequence, it follows that for any prime p and a finite p-group of co-class 1 of order p(n) and exponent p(e), there are at the most seven genus spectra despite the infinite growth of their isomorphism types along with (n, e).