  For a finite group G, we denote by $\mu(G)$, and c(G), the minimal degree of
faithful permutation representation of G, and the minimal degree of faithful
representation of G by quasi-permutation matrices over the complex field C,
respectively. In this article, we study $\mu(G)$, and c(G) for various classes
of finite non-abelian p-groups with cyclic center. We prove a result for
normally monomial p-groups with cyclic center which generalizes a result of
Behravesh for finite p-groups of nilpotency class 2 with cyclic center [5,
Theorem 4.12]. We also compute minimal degrees for some classes of metabelian
p-groups.
