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 Aut(Gamma) congruent to G. In this paper, we prove that alpha(G) <= vertical bar G vertical bar, 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) > vertical bar G vertical bar where the value alpha(G) was previously unknown.