On the derived category of the classical Godeaux surface

We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z/5Z-quotient of the Fermat quintic surface in P3. This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z11⊕Z/5Z. 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 E8. We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence.