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We study the (bigraded) homology of the universal Steenrod algebra Q over the prime field ₂, and we compute the groups $H_{s,s}\left(Q\right)$, s ≥ 0, using some ideas and techniques of Koszul algebras developed by S. Priddy in [5], although we presently do not know whether or not Q is a Koszul algebra. We also provide an explicit formula for the coalgebra structure of the diagonal homology $D⁎\left(Q\right)={⨁}_{s\ge 0}{H}_{s,s}\left(Q\right)$ and show that D⁎(Q) is isomorphic to the coalgebra of invariants Γ introduced by W. Singer in [6].