For the symmetric group $\Sigma_n$, you can take $$ E\Sigma_n = \{\text{injective functions} \{1,\dotsc,n\}\to\mathbb{R}^\infty \}$$ $$ B\Sigma_n = \{\text{subsets of size $n$ in } \mathbb{R}^\infty \}$$
Neil Strickland
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