Smooth rational surfaces of $d=11$ and $\pi=8$ in $\mathbb{P}^5$

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.