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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.

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Communicated by E. ZelmanovIn this paper we take a significant step forward in the classification of 3-groups of coclass 2. Several new phenomena arise. Theoretical and computational tools have been developed to deal with them. We identify and are able to classify an important subset of the 3-groups of coclass 2. With this classification and further extensive computations, it is possible to predict the full classification. On the basis of the work here and earlier work on the p-groups of coclass 1, we formulate another general coclass conjecture. It implies that, given a prime p and a positive integer r, a finite computation suffices to determine the p-groups of coclass r.