Explicit Realization of PSU₃(8) as a Galois Group over ℚ

This problem asks for an explicit realization of the projective special unitary group $\mathrm{PSU}_3(8)$ as a Galois group over $\mathbb{Q}$. The goal is to construct a separable polynomial $f(x)\in\mathbb{Q}[x]$ whose splitting field has Galois group isomorphic to $\mathrm{PSU}_3(8)$, together with a rigorous proof identifying the group. A stronger solution would construct a regular Galois extension of $\mathbb{Q}(t)$ with this group and exhibit a rational specialization that preserves the full Galois group.