of nonisomorphic abelian groups of orders n = 1, 2, ... are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, ... For labeled abelian groups, see . List of small non-abelian groups The numbers of non-abelian groups, by order, are counted by . However, many orders have no non-abelian groups. The orders for which a non-abelian group exists are 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, ... Classifying groups of small order Small groups of prime power order pn are given as follows: Order p: The only group is cyclic. Order p2: There are just two groups, both abelian. Order p3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p2 by a cyclic group of order p. The other is the quaternion group for and a group of exponent p for . Order p4: The classification is complicated, and gets much harder as the exponent of p increases. Most groups of small order have a Sylow p subgroup P with a normal p-complement N for some prime p dividing the order, so can be classified in terms of the possible primes p, p-groups P, groups N, and actions of P on N. In some sense this reduces the classification of these groups to the classification of p-groups. Some of the small groups that do not have a normal p-complement include: Order 24: The symmetric group S4 Order 48: The binary octahedral group and the product Order 60: The alternating group A5. The smallest order for which it is not known how many nonisomorphic groups there are is 2048 = 211. Small Groups Library The GAP computer algebra system contains a package called the "Small Groups library," which provides access to descriptions of small order groups. The groups are listed up to isomorphism. At present, the library contains the following groups: those of order at most 2000 except for order 1024 ( groups in the library; the ones of order 1024 had to be skipped, as there are an additional nonisomorphic 2-groups of order 1024); those of cubefree order at most 50000 (395 703 groups); those of squarefree order; those of order pn