  We construct a linearly normal smooth rational surface S of degree 11 and
sectional genus 8 in the projective fivespace. Surfaces satisfying these
numerical invariants are special, in the sense that $h^1(\mathscr{O}_S(1))>0$.
Our construction is done via linear systems and we describe the configuration
of points blown up in the projective plane. We also present a short list,
generated by the adjunction mapping, of linear systems whom are the only
possibilities for other families of surfaces with the prescribed numerical
invariants.
