Using pairing in Atiyah-Hirzebruch spectral sequence one can show that homology of $BU(n)$ is a free abelian group with basis $\alpha_{k_1}\cdots\alpha_{k_t}$, $k\leqslant n$, where $\alpha_{i} = \big((c_1)^i\big)^* \in H_{2i}(\mathbb C P^\infty)$. Therefore, passing to (co)limits and applying the Thom isomorphism, we have \begin{equation} H_*(MU) \cong \mathbb Z[a_1, a_2, \ldots]. \end{equation}

Since polynomial generators come as image of homology classes of $\mathbb C P^\infty$, $\mathfrak A_p^*$-comodule structure of $H_*(MU; \mathbb F_p)$ can be deduced from the one of $H_*(\mathbb C P^\infty; \mathbb F_p)$.

Is there any similar description of $H_*(MSU; \mathbb F_p)$? Namely, is there any description of $H_*(MSU; \mathbb F_p)$ as an $\mathbb F_p$-algebra and $\mathfrak A_p^*$-comodule?