The class of a nilpotent group refers to the length of a central series, e.g. Hall's lower and upper central series, Higman's Phi-series, the Jenning's series, etc. We develop a refinement of these series which can produce a characteristic series of exponentially longer length. The series retains many familiar properties of the familiar central series, it clarifies the structure of the automorphism groups of various groups, and can be computed in polynomial time. The refinement depends on a Galois connection between tensor products and rings and so far has no visible relationship to the traditional method of selecting certain verbal and marginal subgroups.