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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)" 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).