Summary: A data structure is presented which enables an arbitrary permutation group of degree n to be represented in \(O(n^ 2)\) space. An algorithm is provided which, given a permutation group specified in the usual way as a set of generators, constructs the proposed representation in time \(O(n^ 5)\). The data structure supports fast membership testing, and is more economical than those previously suggested, both in terms of its size and the time required for its initialisation. Essential use is made of the proposed data structure in an efficient algorithm for generating systems of coset representatives; this algorithm may be used to solve certain instances of the so-called ''isomorph rejection'' problem.