  The main goal of this paper is to compute $\mu(\g)$ and $\mu_{nil}(\g)$ for
each nilpotent Lie algebra $\g$ of dimension 6 over a field of characteristic
zero $\k$. Here $\mu(\g)$ and $\mu_{nil}(\g)$ is the minimal dimension of a
faithful representation of $\g$ and the minimal dimension of a faithful
nilrepresentation of $\g$, respectively. We also give a minimal faithful
representation and nilrepresentation for each nilpotent Lie algebra of
dimension $\leq 6$.
