  Using numerical arguments we find that for $N$ = 306 a tetrahedral
configuration ($T_h$) and for N=542 a dihedral configuration ($D_5$) are likely
the global energy minimum for Thomson's problem of minimizing the energy of $N$
unit charges on the surface of a unit conducting sphere. These would be the
largest $N$ by far, outside of the icosadeltahedral series, for which a global
minimum for Thomson's problem is known. We also note that the current
theoretical understanding of Thomson's problem does not rule out a symmetric
configuration as the global minima for N=306 and 542. We explicitly find that
analogues of the tetrahedral and dihedral configurations for $N$ larger than
306 and 542, respectively, are not global minima, thus helping to confirm the
theory of Dodgson and Moore (Phys. Rev. B 55, 3816 (1997)) that as $N$ grows
dislocation defects can lower the lattice strain of symmetric configurations
and concomitantly the energy. As well, making explicit previous work by
ourselves and others, for $N<1000$ we give a full accounting of
icosadeltahedral configuration which are not global minima and those which
appear to be, and discuss how this listing and our results for the tetahedral
and dihedral configurations may be used to refine theoretical understanding of
Thomson's problem.
