In this article, we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex d-dimensional family of compact Riemann surfaces, where d >= 1. We provide conditions under which these bounds are sharp. In addition, for the one-dimensional case, we construct and describe an explicit family attaining the bound for infinitely many genera. We obtain similar results for the case of p-groups of automorphisms.