  In previous work we showed that the Hurwitz space of W(E_6)-covers of the
projective line branched over 24 points dominates via the Prym-Tyurin map the
moduli space A_6 of principally polarized abelian 6-folds. Here we determine
the 25 Hodge classes on the Hurwitz space of W(E_6)-covers corresponding to the
25 irreducible representations of the Weyl group W(E_6). This result has direct
implications to the intersection theory of the toroidal compactification A_6.
In the final part of the paper, we present an alternative, elementary proof of
our uniformization result on A_6 via Prym-Tyurin varieties of type W(E_6).
