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

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 greater than or equal to 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. (C) 2003 Elsevier Inc. All rights reserved.