The Computational Algebra Group has been in existence since the early 1970's, when John Cannon and Greg Butler developed many of the standard permutation group algorithms. Starting in the late 1970's, in joint work with Charles Leedham-Green, the group commenced the design of algorithms for p-groups and soluble groups.

In the early 1970's, in conjunction with Joachim Neubüser's group in Aachen, the Computational Algebra Group developed the first publicly available software package for group theory: Group. In 1975 it commenced the development of the Cayley system for group theory and related areas. Following its release in 1982, Cayley has been successfully applied to a very wide variety of problems both within and outside mathematics.

A great deal of experience with algebraic structural computation was gained from Cayley in the context of computing structural information about groups. Consequently a completely new system, Magma emerged. Distribution of Magma V1 commenced at the end of 1993. To date, Magma has been cited in over 2,500 research publications.