A Classification of Metacyclic 2-Groups

A finite p-group G is called metacyclic if it has a cyclic normal subgroup A such that G\A is cyclic. In 1973, King classified metacyclic p-groups; in 1987, Newman and the first author developed a new approach to metacyclic p-groups suggested by the p-group generation algorithm, they found new presentations for these groups. However, their results were only announced in the case of p odd, and no proof was given. The purpose of this paper is to give a new proof of their results in the case of p=2, which is independent of the p-group generation algorithm.