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Given a finite dimensional Lie algebra $\mathfrak{g}$, let$\mathfrak{z}(\mathfrak{g})$ denote the center of $\mathfrak{g}$ and let$\mu(\mathfrak{g})$ be the minimal possible dimension for a faithfulrepresentation of $\mathfrak{g}$. In this paper we obtain$\mu(\mathcal{L}_{r,2})$, where $\mathcal{L}_{r,k}$ is the free $k$-stepnilpotent Lie algebra of rank $r$. In particular we prove that$\mu(\mathcal{L}_{r,2})= \left\lceil \sqrt{2r(r-1)} \right\rceil + 2$ for $r\geq 4$. It turns out that $\mu(\mathcal{L}_{r,2})\sim\mu\big(\mathfrak{z}(\mathcal{L}_{r,2})\big) \sim2\sqrt{\dim\mathcal{L}_{r,2}} $ (as $r\to\infty$) and we present some evidencethat this could be true for $\mathcal{L}_{r,k}$ for any $k$, this isconsiderably lower than the known bounds for $\mu(\mathcal{L}_{r,k})$, whichare (for fixed $k$) polynomial in $\dim\mathcal{L}_{r,k}$.

Minimal faithful representations of the free 2-step nilpotent Lie algebra of the rank $r$