Let $G$ be a group and let $\Gamma\_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}\_G(k)=\Gamma\_G(k)/\Gamma\_G(k+1)$ and $$\mathcal{L}\_G:=\bigoplus\_{k\geq 1}\mathcal{L}\_G(k).$$ Then $\mathcal{L}\_G$ has a graded Lie algebra structure induced from the commutator bracket on $G$.


My question:


To what extent can one determine $G$, if $\mathcal{L}\_G$ is known? It is true that $G$ is uniquely determined by $\mathcal{L}\_G$? If yes, how to find $G$, knowing $\mathcal{L}\_G$?



Does the associated Lie algebra determine a group?
It seems to me that even $Q\_8$ and $D\_8$ have isomorphic algebras. In fact, one can construct lots of similar pairs of $2$-nilpotent $2$-groups: roughly, the Carnot algebra captures a symplectic form only (which are all isomorphic), whereas the group itself depends on a quadratic extension of this form, and those differ by the $\operatorname{Arf}$-invariant.


