$\mathrm {SL}(2, 5)$ Has No Smooth Effective One-fixed-point Action on $S^8$

We prove that there is no effective smooth one-fixed-point action of $\mathrm {SL}(2,5)$, the special linear group of $2\times 2$ matrices over $\mathbb {Z}_5$, on the $8$-dimensional sphere. The method of proof involves the intersection form.