I don't think the cohomology is known. As an indication of this I would refer to the 2007 paper *"[On the Stiefel–Whitney classes of the representations associated with Spin(15)](http://dx.doi.org/10.2140/gtm.2007.11.141)"* by Mamoru Mimura and Tetsu Nishimoto (see https://msp.org/gtm/2007/11/p007.xhtml). There they state that *"calculating $H^\ast(BSs(16);Z/2Z)$ 
seems to be as difficult
as calculating $H^\ast(BE_8;Z/2Z)$"* (which is also still unknown).