  Consider a cubic surface satisfying the mild condition that it may be
described in Sylvester's pentahedral form. There is a well-known Enriques or
Coble surface S with K3 cover birationally isomorphic to the Hessian surface of
this cubic surface. We describe the nef cone and the (-2)-curves of S. In the
case of pentahedral parameters (1, 1, 1, 1, nonzero t) we compute the
automorphism group of S. For t not 1 it is the semidirect product of the free
product (Z/2)*(Z/2)*(Z/2)*(Z/2) by the symmetric group S4. In the special case
t=1/16 we study the action of Aut(S) on an invariant smooth rational curve C on
the Coble surface S. We describe the action and its image, both geometrically
and arithmetically. In particular, we prove that Aut(S)-->Aut(C) is injective
in characteristic 0 and we identify its image with the subgroup of PGL2 coming
from the symmetries of a regular tetrahedron and the reflections across its
facets.

Algebraic Geometry