Smallest graphs with given automorphism group

For a finite group G , denote by α (G) the minimum number of vertices of any graph Γ having Aut(Γ )≅ G . In this paper, we prove that α (G)≤ |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.