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