On the Hurewicz image of the Steinberg summand M(n)

Given a prime p, we define a spectrum M(n) by the stable summand of the classifying space Bn(Z/p) induced by the Steinberg idempotent. We show that the mod p Hurewicz map π∗(M(n))→H∗(M(n);Fp) has trivial image in most dimensions.