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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    $\begingroup$ Very interesting question. I too would like to know answers . A rough attempt: Take an element of the group whose conjugacy class is 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
  • $\begingroup$ Also, have a look at : link.springer.com/article/10.1007/BF02261693 $\endgroup$ – Geoff Robinson Oct 24 '16 at 23:33
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    $\begingroup$ @GeoffRobinson, you mean David Easdown I believe. $\endgroup$ – Benjamin Steinberg Oct 25 '16 at 0:03
  • $\begingroup$ @BenjaminSteinberg :Yes, thanks, indeed I do. $\endgroup$ – Geoff Robinson Oct 25 '16 at 0:09
  • $\begingroup$ Correction: In general, I don't think this is an easy problem: you might check out papers by D. Easdown and N. Saunders on Arxiv for some references. $\endgroup$ – Geoff Robinson Oct 25 '16 at 0:10