  We study the moduli space $\widetilde{\calM}$ of marked cubic surfaces. By
classical work of A.\,B. Coble, this has a compactification $\widetilde{M}$,
which is linearly acted upon by the group $W(E_6)$. $\widetilde{M}$ is given as
the intersection of 30 cubics in $\bP^9$. For the morphism $\widetilde{calM}
\to \bP(1,2,3,4,5)$ forgetting the marking, followed by Clebsch's invariant
map, we give explicit formulas. I.e., Clebsch's invariants are expressed in
terms of Coble's irrational invariants. As an application, we give an
affirmative answer to the inverse Galois problem for cubic surfaces over
$\bbQ$.
