  We extend the notion of a fundamental negatively $\mathbb Z$-graded Lie
algebra $\mathfrak{m}_x=\bigoplus_{p\leq -1}\mathfrak{m}_x^p$ associated to any
point of a Levi nondegenerate CR manifold to the class of $k$-nondegenerate CR
manifolds $(M,\mathcal D,\mathcal J)$ for all $k\geq 2$ and call this invariant
the core at $x\in M$. It consists of a $\mathbb Z$-graded vector space
$\mathfrak{m}_x=\bigoplus_{p\leq k-2}\mathfrak{m}_x^p$ of height $k-2$ endowed
with the natural algebraic structure induced by the Tanaka and Freeman
sequences of $(M,\mathcal D,\mathcal J)$ and the Levi forms of higher order. In
the case of CR manifolds of hypersurface type we propose a definition of a
homogeneous model of type $\mathfrak m$, that is, a homogeneous
$k$-nondegenerate CR manifold $M=G/G_o$ with core $\mathfrak m$ associated with
an appropriate $\mathbb Z$-graded Lie algebra $Lie(G)=\mathfrak
g=\bigoplus\mathfrak g^p$ and subalgebra $Lie(G_o)=\mathfrak
g_o=\bigoplus\mathfrak g_o^p$ of the nonnegative part $\bigoplus_{p\geq
0}\mathfrak g^p$. It generalizes the classical notion of Tanaka of homogeneous
model for Levi nondegenerate CR manifolds and the tube over the future light
cone, the unique (up to local CR diffeomorphisms) maximally homogeneous
$5$-dimensional $2$-nondegenerate CR manifold. We investigate the basic
properties of cores and models and study the $7$-dimensional CR manifolds of
hypersurface type from this perspective. We first classify cores of
$7$-dimensional $2$-nondegenerate CR manifolds up to isomorphism and then
construct homogeneous models for seven of these classes. We finally show that
there exists a unique core and homogeneous model in the $3$-nondegenerate
class.
