Geodesic
On the Minimum Order of Graphs with Given Group
Canadian mathematical bulletin, Tome 17 (1974) no. 4, pp. 467-470

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For G a, finite group let α(G) denote the minimum number of vertices of the graphs X the automorphism group A(X) of which is isomorphic to G.G. Sabidussi proved [1], that α(G)=0(n log d) where n=\G\ and d is the minimum number of generators of G.As 0(log n) is the best possible upper bound for d, the result established in [1] implies that α(G)=0(n log log n). 
Babai, László. On the Minimum Order of Graphs with Given Group. Canadian mathematical bulletin, Tome 17 (1974) no. 4, pp. 467-470. doi: 10.4153/CMB-1974-082-9
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     title = {On the {Minimum} {Order} of {Graphs} with {Given} {Group}},
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     year = {1974},
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     number = {4},
     doi = {10.4153/CMB-1974-082-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1974-082-9/}
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