Groups and nilpotent Lie rings whose order is the sixth power of a prime

We prove that there are 3p2+39p+344+24gcd(p−1,3)+11gcd(p−1,4)+2gcd(p−1,5) isomorphism types of groups and nilpotent Lie rings with order p6 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 p6 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.