  Due to the deep work of Tillmann, Madsen, Weiss and Galatius, the cohomology
of the stable mapping class group $\gaminf$ is known with rational or finite
field coefficients. Little is known about the integral cohomology. In this
paper, we study the first four cohomology groups. Also, we compute the first
few steps of the Postnikov tower of $B \gaminf^+$, the Quillen plus
construction applied to $B \gaminf$. Our method relies on the Madsen-Weiss
theorem, a few known computations of stable homotopy groups of spheres and
projective spaces and on a certain action of the binary icosahedral group on a
surface. Using the latter, we can also describe an explicit geometric generator
of the third homotopy group $\pi_3 (B \gaminf)$.
