Non-trivial blocks of order $16$ of $D_{16}$ acting on $8$-gon vertices
Solution 1:
Hint. The action of $D_{16}$ on the set $X$ of vertices is clearly transitive, so the blocks containing a fixed vertice $x\in X$ are in one-to-one correspondence with the subgroups of $D_{16}$ containing the stabilizer $G_x$ of $x$.
Now , what are the rotations fixing $x$ ? what are the symmetries fixing $x$?