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No abstract

We prove that there are 3p 2 + 39p + 344 + 24 gcd(p − 1, 3) + 11 gcd(p − 1, 4) + 2 gcd(p − 1, 5) isomorphism types of groups and nilpotent Lie rings with order p 6 for every prime p 5. We establish the result, and power-commutator presentations for the groups, in various ways. The most novel method constructs product presentations for nilpotent Lie rings with order p 6 and then uses the Baker-Campbell-Hausdorff formula to construct power-commutator presentations for the corresponding groups. Public access to the group presentations is provided via a database distributed with computer algebra systems.