For a finite group G, denote by α(G) the minimum number of vertices of any graph having Aut() ∼= G. In this paper, we prove that α(G) ≤ |G|, with specifiedexceptions. The exceptions include four infinite families of groups, and 17 other smallgroups. Additionally, we compute α(G) for the groups G such that α(G) > |G| wherethe value α(G) was previously unknown.