A CAT(0) cube complex is a high-dimensional generalization of a tree. A group G has property FC_n if every action of G on an n-dimensional cube complex has a global fixed point. Thus the properties FC_n give a stratification between property FA (=FC_1) and property (T). Random groups in Gromov’s density model are known to have FA at all densities and (T) for densities greater than 1/3. We will prove that random groups have FC_n for every n at every density.