On the Hurewicz image of the Steinberg summand

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