  We construct an exceptional sequence of length 11 on the classical Godeaux
surface X which is the Z/5-quotient of the Fermat quintic surface in P^3. This
is the maximal possible length of such a sequence on this surface which has
Grothendieck group Z^11+Z/5. In particular, the result answers Kuznetsov's
Nonvanishing Conjecture, which concerns Hochschild homology of an admissible
subcategory, in the negative. The sequence carries a symmetry when interpreted
in terms of the root lattice of the simple Lie algebra of type E_8. We also
produce explicit nonzero objects in the (right) orthogonal to the exceptional
sequence.
