Goal: For a given positive integer n, list the groups of order n, verifying that no two of the groups listed are isomorphic. For each group, we provide its order profile and some of the following: Cayley digraph; lattice (or list) of subgroups; conjugacy classes; automorphism group; inner automorphism group; and alternate representations. For some small values of n, we outline a proof that the list is complete.
Hot off the Press: Groups of Order 72. There are 50 of them, forty-four of which are nonabelian. Surprisingly (to me, at least), no two of them have the same order profile, making it relatively easy to verify that no two of them are isomorphic. (I wonder, How do we know the list is complete? Someday - perhaps when I retire - I hope to understand how one goes about proving that there are exactly 50 groups of order 72, up to isomorphism.)
Even Hotter off the Press: Groups of Order 64. There are 267 of them, 256 of which are nonabelian. Part I covers the eleven abelian groups, the three "special" nonabelian groups, and the 56 nonabelian groups that can be expressed as a direct product of smaller groups. Part II covers an additional 85 nonabelian groups that can be expressed as a semi-direct product of two abelian groups. Part III presents 95 nonabelian groups that can be expressed as the semi-direct product of two groups, at least one of which is nonabelian. while Part IV covers the seventeen nonabelian groups that can't be expressed as a semi-direct product of two smaller groups.
Under Construction: Groups of Order 96: There are 231 of them, 224 of which are nonabelian.