jmc

We want an integer $n$ such that $8<\sqrt[3]{n}<8.1$. Cubing each part of the inequality gives $8^3<n<8.1^3$, or $512<n<531.441$. We know $n$ is a multiple of 18, so we try to find a multiple of 18 in this range (we can do this by letting $n=18k$, and trying out different integer values of $k$). We find that $18\cdot 29=522$ is the only multiple of 18 in this range. So $\boxed{522}$ is the answer.