This paper proves a lower bound on the index of a maximal abelian subgroup A in a p-group G, in terms of the maximum size of a conjugacy class intersecting A, and the maximum size of the center of the centralizer of an element in A. The main result shows this index is at least n/(b+l) where n is the exponent on |G:Z(G)|, b bounds conjugacy class sizes, and l bounds size...