The S = 11 ground states of the Mn-7 family of mixed-valence complexes with a metal-centered hexagonal topology have been found by density functional theory calculations to arise by spin frustration involving small differences in the magnitudes of the two weakest interactions controlling the alignment of the central spin. Targeted structural perturbation has allowed a complex with the central spin flipped to be discovered, which thus possesses the maximum S = 16 ground state.