We survey the problem of constructing the groups of a given finite order. We provide an extensive bibliography and outline practical algorithmic solutions to the problem. Motivated by the millennium, we used these methods to construct the groups of order at most 2000; we report on this calculation and describe the resulting group library.

We introduce three practical algorithms to construct certain finite groups up to isomorphism. The first one can be used to construct all soluble groups of a given order. This method can be restricted to compute the soluble groups with certain properties such as nilpotent, non-nilpotent or supersoluble groups. The second algorithm can be used to determine the groups of order p n · q with a normal Sylow subgroup for distinct primes p and q. The third method is a general method to construct finite groups which we use to compute insoluble groups.

Recently, we developed practical algorithms to determine up to isomorphism the groups of a given order. Here we describe details on the implementations and the applications of these methods. In particular, we report on the determination of the groups of order at most 1000 except 512 and 768.

Abstract. We announce the construction up to isomorphism of the 49 910 529 484 groups of order at most 2000.

We describe a method to determine up to isomorphism the groups of order q n · p for a fixed prime-power q n and indeterminate prime p = q. We report on the explicit construction of all groups of order 2 n · p for n ≤ 8 and 3 n · p for n ≤ 6. In particular, we show that there are 1 090 235 groups of order 768.