  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 $d \geqslant 1.$ 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.
