The following question arises while I am reading a paper of [B. N. Cooperstein.](https://link.springer.com/article/10.1007/BF02761072)


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? 




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**Edit:** A table for permutation degree of simple groups can be found [here](https://www.math.wisc.edu/~grizzard/papers/table.pdf).



On some classical groups with small permutation degree
For me the article by B. N. Cooperstein is behind a paywall. 
But I think the object you are looking for is the Hoffman–Singleton graph. There is a wikipedia page about that graph, see


<https://en.wikipedia.org/wiki/Hoffman%E2%80%93Singleton_graph>


