  We study the web of correspondences linking the exceptional Lie algebras
$E_{8,7,6}$ and the sporadic simple groups Monster, Baby and the largest
Fischer group. This is done via the investigation of classical enumerative
problems on del Pezzo surfaces in relation to the cusps of certain subgroups of
$PSL(2,R)$ for the relevant McKay-Thompson series in Generalized Moonshine. We
also study Conway's sporadic group, as well as its association with the
Horrocks-Mumford bundle.
