Let $G$ be a finite group. The faithful dimension of $G$ is defined to be the smallest possible dimension for a faithful complex representation of $G$. Aside from its intrinsic interest, the problem of determining the faithful dimension of $p$-groups is motivated by its connection to the theory of essential dimension. In this talk, we will address this problem for groups of the form $\mathbf{G}_p:=\exp(\mathfrak{g} \otimes_{\mathbb{Z}}\mathbb{F}_p)$, where $\mathfrak{g}$ is a nilpotent $\mathbb{Z}$-Lie algebra of finite rank, and $\mathbf{G}_p$ is the $p$-group associated to $\mathfrak{g} \otimes_{\mathbb{Z}}\mathbb{F}_p$ in the Lazard correspondence. We will show that in general the faithful dimension of $\mathbf{G}_p$ is given by a finite set of polynomials associated to a partition of the set of prime numbers into Frobenius sets. At the same time, we will show that for many naturally arising groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single polynomial. The arguments are reliant on various tools from number theory, model theory, combinatorics and Lie theory.