Using an efficient numerical scheme that exploits spatial symmetries and spin parity, we have obtained the exact low-lying eigenstates of exchange Hamiltonians for the high-nuclearity spin clusters, Mn-12, Fe-8, and V-15. The largest calculation involves the Mn-12 cluster that spans a Fock space of a hundred million. Our results show that the earlier estimates of the exchange constants need to be revised for the Mn-12 cluster to explain the level ordering of low-lying eigenstates. In the case of the Fes cluster, correct level ordering can be obtained, which is consistent with the exchange constants for the already known clusters with butterfly structure. In the V-15 cluster, we obtain an effective Hamiltonian that reproduces exactly, the eight low-lying eigenvalues of the full Hamiltonian.