We prove a theorem with the following corollary: For each integer k greater than or equal to 1, an arbitrary finite group G embeds into some finite group G(k) for which there exists an Eilenberg-Mac Lane CW-space X = K(G(k),1) whose finite n-skeleton X-n has Euler-Poincare characteristic chi (X-n) = 1 + (-1)(n)dH(n)(Gk) for all n less than or equal to k. The theorem can be viewed as a generalisation of a result of J. Harlander [1996, J. Algebra 182, 511-521] on the embedding of finite groups into groups with "efficient" presentations. (C) 2001 Academic Press.