For a finite group $G$, denote by $α(G)$ the minimum number of vertices of any graph $Γ$ having $\text{Aut}(Γ)\cong G$. In this paper, we prove that $α(G)\leq |G|$, with specified exceptions. The exceptions include four infinite families of groups, and 17 other small groups. Additionally, we compute $α(G)$ for the groups $G$ such that $α(G)> |G|$ where the value $α(G)$ was previously unknown.