We use the linear threshold model to study the diffusion of information on a network generated by the stochastic block model. We focus our analysis on a two-community structure where the initial set of informed nodes lies only in one of the two communities and we look for optimal network structures, i.e., those maximizing the asymptotic extent of the diffusion. We find that, constraining the mean degree and the fraction of initially informed nodes, the optimal structure can be assortative (modular), core-periphery, or even disassortative. We then look for minimal cost structures, i.e., those for which a minimal fraction of initially informed nodes is needed to trigger a global cascade. We find that the optimal networks are assortative but with a structure very close to a core-periphery graph, i.e., a very dense community linked to a much more sparsely connected periphery.