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According to the wikipedia page, every group is indeed the automorphism group of some graph. This was proven independently by in

de Groot, J. (1959) 1959), Groups represented by homeomorphism groups, Mathematische Annalen 138

and

Sabidussi, Gert (1960).1960), Graphs with given infinite group, Monatshefte für Mathematik 64: 64–67.

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According to the wikipedia page, every group is indeed the automorphism group of some graph. This was proven independently by de Groot (1959) and Sabidussi (1960).