Covering a Polish group by translates of a nowhere dense set

We show that, for every nonlocally compact Polish group G with a left-invariant complete metric ρ, we have covG=cov(M). Here, covG is the minimal number of translates of a fixed closed nowhere dense subset of G, which is needed to cover G, and cov(M) is the minimal cardinality of a cover of the real line R by meagre sets.