Suppose I have a finite group $G.$ How hard is it to find the (a?) minimal degree permutation representation of $G?$ The second part of the question is: is there a table of such (hopefully for parametrized families)?

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Suppose I have a finite group $G.$ How hard is it to find the (a?) minimal degree permutation representation of $G?$ The second part of the question is: is there a table of such (hopefully for parametrized families)?

small. The action by inner automorphisms on that class might be (should avoid central elements for obvious reasons) one possible candidate for answer. $\endgroup$ – P Vanchinathan Oct 24 '16 at 22:26