The following question arises while I am reading a paper of B. N. Cooperstein.

In the Table 1 of his paper, two groups have smaller permutation degree compared to other members in their family: $\mathrm{PSp}_4(3)$ with permutation degree 27 and $\mathrm{PSU}_3(5)$ with permutation degree 50, while $\mathrm{PSp}_{2n}(q)$ generally has a permutation degree of $$\frac{q^{2n}-1}{q-1}$$ and $\mathrm{PSU}_3(q)$ has a permutation degree of $q^3+1$. It is already known to Camille Jordan that $\mathrm{PSp}_4(3)$ is a permutation group on 27 lines of a cubic surface. I am curious if there is a similar geometric configuration for $\mathrm{PSU}_3(5)$.

**Question:** Is there any neat geometric configuration (with 50 elements) that the group $\mathrm{PSU}_3(5)$ is a permutation group which acts faithfully on the configuration?

**Edit:** A table for permutation degree of simple groups can be found here.