We show obstructions to the existence of a coclosed $G_2$-structure on a Lie algebra $mathfrak g$ of dimension seven with non-trivial center. In particular, we prove that if there exist a Lie algebra epimorphism from $mathfrak g$ to a six-dimensional Lie algebra $mathfrak h$, with kernel contained in the center of $mathfrak g$, then any coclosed $G_2$-structure on $mathfrak g$ induces a closed and stable three form on $mathfrak h$ that defines an almost complex structure on $mathfrak h$. As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed $G_2$-structures. We also prove that each one of these Lie algebras has a coclosed $G_2$-structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed $G_2$-structures. The existence of contact metric structures is also studied.
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