Magnetic nanoparticle clusters have several biomedical and engineering applications, and revealing the basic interplay between particle configuration and magnetic properties is important for tuning the clusters for specific uses. Here, we consider the nanoparticles as macrospins and use computer simulations to determine their magnetic configuration when placed at the vertices of various polyhedra. We find that magnetic dipoles of equal magnitude arrange in flux-closed vortices on a layer basis, giving the structures a null remanent magnetic moment. Assigning a toroidal moment to each layer, we find that the geometrical arrangement, i.e., “triangular packing” vs. “square packing,” of the moments in the adjacent layer determines whether the flux-closed layers are ferrotoroidal (co-rotating vortices) or antiferrotoroidal (counter-rotating vortices). Interestingly, upon adding a single magnetic moment at the center of the polyhedra, the central moment relaxes along one of the principal axes and induces partial alignment of the surrounding moments. The resulting net moment is up to nearly four times that of the single moment added. Furthermore, we model quasi-static hysteresis loops for structures with and without a central moment. We find that a central moment ensures an opening of the hysteresis loop, and the resultant loop areas are typically many-fold larger compared to the same structure without a central moment.