Let G be a finite group. Define α(G) as the minimum number of vertices among all graphs Γ such that AutΓ ∼ = G. For any prime p, all p-groups of order pn having cyclic subgroups of order pn-1 have been completely classified. Here, we consider one family of groups called modular p-groups, denoted by Modn(p), for an odd prime p and n ≥ 3. We compute the order of vertex-minimal graphs with Modn(p)-symmetry. The fixing number of a graph Γ is defined as the smallest number of vertices in V (Γ) that, when fixed, makes AutΓ trivial. For a finite group G, the fixing set is defined as the set of all fixing numbers of graphs having automorphism groups isomorphic to G. We show that any graph Γ whose automorphism group is a modular p-group has the fixing number 1. As a result, the modular p-group’s fixing set becomes {1}. Keywords: Automorphism group, p-group, vertex-minimal graph, fixing number, fixing set.