V which take each Vi to Vi+m form a subgroup of Aut(V) denoted Um. If V is a vector space over Z/pZ, then U1 is a Sylow p-subgroup of Aut(V) = GL(n, p), and the terms of its lower central series are just the Um. In terms of matrices, Um are those upper triangular matrices with 1s one the diagonal and 0s on the first m−1 superdiagonals. The group U1 has order pn·(n−1)/2, nilpotency class n, and exponent pk where k is the least integer at least as large as the base p logarithm of n. Classification The groups of order pn for 0 ≤ n ≤ 4 were classified early in the history of group theory, and modern work has extended these classifications to groups whose order divides p7, though the sheer number of families of such groups grows so quickly that further classifications along these lines are judged difficult for the human mind to comprehend. For example, Marshall Hall Jr. and James K. Senior classified groups of order 2n for n ≤ 6 in 1964. Rather than classify the groups by order, Philip Hall proposed using a notion of isoclinism of groups which gathered finite p-groups into families based on large quotient and subgroups. An entirely different method classifies finite p-groups by their coclass, that is, the difference between their composition length and their nilpotency class. The so-called coclass conjectures described the set of all finite p-groups of fixed coclass as perturbations of finitely many pro-p groups. The coclass conjectures were proven in the 1980s using techniques related to Lie algebras and powerful p-groups. The final proofs of the coclass theorems are due to A. Shalev and independently to C. R. Leedham-Green, both in 1994. They admit a classification of finite p-groups in directed coclass graphs consisting of only finitely many coclass trees whose (infinitely many) members are characterized by finitely many parametrized presentations. Every group of order p5 is metabelian. Up to p3 The trivial group is the only group of order one, and the cyclic group Cp is the only group of order p. There are exactly two groups of order p2, both abelian, namely Cp2 and Cp × Cp. For example, the cyclic group C4 and the Klein four-group V4 which is C2 × C2 are both 2-groups of order 4. There are three abelian groups of order p3, namely Cp3, Cp2 × Cp, and