We have computed the optimized geometries and energy separations of low-lying electronic states of the lead hexamer (Pb-6) and its positive and negative ions. Our techniques have included high level relativistic electron correlation techniques such as complete active space multiconfiguration self-consistent field (CAS-MCSCF) method followed by large scale multireference singles plus doubles configuration interaction (MRSDCI) and relativistic configuration interaction (RCI) computations that included up to 16 million configurations. Our computed results have facilitated the assignment of the anion photodetachment spectra of Pb-6(-) and also in the prediction of the properties of yet to be observed electronic states. A (1)A(1g) tetragonal bipyramid structure (D-4h symmetry) is found as the ground state for Pb-6. The excitation energy, atomization energies, ionization potentials, and vertical and adiabatic electron affinities are computed and compared with the experimental results. We have assigned the observed X, A, B, C, D, and E states of the anion photoelectron spectra of Pb-6(-), and discuss spin-orbit versus Jahn-Teller effects.