Let $S^0_p$ be the $p$-adic sphere spectrum. Let $GL_1(S^0_p)$ be the unit component of $\Omega^{\infty}S^0_p$. For any map $ X \to BGL_1(S_p^0)$ we get a Thom spectrum call it $Mf$. Now consider the identity map on $GL_1(S^0_p)$, the Thom spectrum associated to that map is what I call the universal Thom spectra and denote it by $\mathcal{M}$.

My question is what are the homotopy groups of this Thom spectrum?

(I suspect that $\pi_0(\mathcal{M}) = \mathbb{Z}/p$ and $\mathcal{M}$ is an $E_{\infty}$ ring spectrum, which may give us some hint.)

Let $S^0_p$ be the $p$-adic sphere spectrum. Let $GL_1(S^0_p)$ be the set of unit componen of $\Omega^{\infty}S^0_p$. For any map $ X \to BGL_1(S_p^0)$ we get a Thom spectrum call it $Mf$. Now consider the identity map on $GL_1(S^0_p)$, the Thom spectrum associated to that map is what I call the universal Thom spectra and denote it by $MGL_1(S^0_p)$.

My question is what are the homotopy groups of this Thom spectrum?