Smallest graphs with given automorphism group

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.