TITLE: Dyer-Lashof operations and the homology of GL_n
QUESTION [10 upvotes]: For any ring R, $\bigsqcup_n {BGL}_n(R)$ is an $E_\infty$-space. Are there examples of rings where people have calculated $H_*(\bigsqcup_n {BGL}_n(R);\mathbb{Z}/2)$ and determined the Dyer-Lashof operations?

REPLY [6 votes]: Taking the ring of real numbers $\mathbb{R}$, complex numbers $\mathbb{C}$, or $\mathbb{H}$, the paper of Priddy 
`DYER-LASHOF OPERATIONS FOR THE CLASSIFYING SPACES OF CERTAIN MATRIX GROUPS' Quart J. Math. Oxford (3), 26 (1975), 179-93
provides example of computations and detailed formula of the sort you may look for, bearing in mind that in these cases the monoid you take, is homotopy equivalent to the one considered in the paper.