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. © 2003 Elsevier Inc. All rights reserved