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We compute a family of coverings with four ramification points, defined over$\mathbb{Q}$, with regular Galois group $PSL_5(2)$.  On the one hand, this is (to my knowledge) the first explicit polynomial withgroup $PSL_5(2)$ over $\mathbb{Q}(t)$. On the other hand, it also positivelyanswers the question whether $PSL_5(2)$ is the monodromy group of a rationalfunction over $\mathbb{Q}$. At least this does not follow from consideringclass triples in $PSL_5(2)$, as there are no rigid, rational genus-zerotriples. Also, for 4-tuples, our family is the only one with a Hurwitz curve ofgenus zero (however it does not seem immediately clear without explicitcomputations whether this curve can be defined as a rational curve over$\mathbb{Q}$). There are also genus zero families with five branch points, andmaybe their Hurwitz spaces can be shown to have rational points; however, sofar I have not seen such arguments.

A family of polynomials with Galois group $PSL_5(2)$ over $\mathbb{Q}(t)$