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

Abstract We prove that there are 3 p 2 +39 p +344+24gcd( p −1,3)+11gcd( p −1,4)+2gcd( 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.