I’m reading Pengelley’s paper “*The mod 2 homology of $MSO$ and $MSU$ as $\mathfrak A^*$ comodule algebras, and the cobordism ring*”.

He has chosen very special generators $z_n \in H_n(MO; \mathbb F_2)$.

If $n = 2^t-1$, then $z_{2^t-1} = \varphi(w_{2^t-1}^*)$, where $\varphi$ is the Thom isomorphism. If $n \neq 2^t -1$, then $z_n$ is the Hurewicz image of a $n$-manifold $M_n$. Here $\{M_n\}_{n \neq 2^t-1}$ is a set of polynomial generators of $\pi_*(MO)$ with the additional properties that $M_{2k-1}$ is an orientable submanifold of $M_{2k}$ dual to $w_1(M_{2k})$ which is itself the reduction of an integral class.

These generators are useful to explicitly describe $\mathfrak A_2^*$ comodule structure of $H_*(MSO;\mathbb F_2)$ using the structure of $H_*(MO; \mathbb F_2)$.

**Here is the question.**
Is there a similar set of polynomial generators $\{M_n\in \pi_{2n}(MU)\}_{n\neq p^t -1}$ ($p$ is odd)
with the property that $M_{2k-1}$ is dual to $c_1(M_{2k})$? Or is there another way to describe the $\mathfrak A_p$-comodule structure of $H_*(MSU; \mathbb F_p)$ ($p$ is odd)?