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Timothy Gowers asked if the isomorphism type of a group might be determined by the multiset of isomorphism types of all k-generated subgroups of some bounded k.  Glauberman and Grabowski independently found the same family of counter-examples showing at least $g>\sqrt{\log |G|}$ is required.  Babai conjectured this lower bound was tight.  We prove the correct tight bound of $k=\log |G|-2$ by creating a family of non-isomorphic groups having all proper subgroups isomorphic, all proper quotient groups isomorphic, isomorphic character tables, and isomorphic automorphism groups.  We further show this family of groups has the maximal number of subgroups, quotients, and characters, of any group of that same order.  Paradoxically, these conditions also prove we can decide isomorphism for these groups polylogarithmic time.