In [2] we studied Milgram's Complex, C(n – 1), which was first defined in [3], in the following manner. Let Sn, the permutation group of n symbols, act on Rn in the obvious manner; put α(x) = (y), where yi = xα–1(i). Let s = (1, 2, …, n), then C(n – 1) is the convex hull of the points α(s), α ∊ Sn. Here we shall generalise this construction as follows. Let G be a subgroup of Sn, and let v ∊ Rn. Then C(G, v) is the convex hull of α(v), α ∊ G. We prove invariance over v subject to certain restrictions, give counter-examples to shew lack of invariance if we alter G, discuss how we may describe C(G, v), shew that the only “nice” case is essentially when G is Sn, and lastly give some examples.