  We give a systematic study of the model theory of generic nilpotent groups
and Lie algebras. We show that the Fra\"iss\'e limit of 2-nilpotent groups of
exponent $p$ studied by Baudisch is 2-dependent and NSOP$_{1}$. We prove that
the class of $c$-nilpotent Lie algebras over an arbitrary field, in a language
with predicates for a Lazard series, is closed under free amalgamation. We show
that for $2 < c$, the generic $c$-nilpotent Lie algebra over $\mathbb{F}_{p}$
is strictly NSOP$_{4}$ and $c$-dependent. Via the Lazard correspondence, we
obtain the same result for $c$-nilpotent groups of exponent $p$, for an odd
prime $p > c$.
