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Is the $E_\infty$-structure on the cochain complex of a $K(G,n)$ readily understandable?

One way to construct an $E_\infty$-algebra is to consider the cochain complex $C^*(X;M)$ for $X$ a topological space and $M$ a module over some ring $\Lambda$. From what I can recall, the $E_\infty$-a …