  In this paper we use the recent developments in the representation theory of
locally compact quantum groups, to assign, to each locally compact quantum
group $\mathbb{G}$, a locally compact group $\tilde \mathbb{G}$ which is the
quantum version of point-masses, and is an invariant for the latter. We show
that "quantum point-masses" can be identified with several other locally
compact groups that can be naturally assigned to the quantum group
$\mathbb{G}$. This assignment preserves compactness as well as discreteness
(hence also finiteness), and for large classes of quantum groups, amenability.
We calculate this invariant for some of the most well-known examples of
non-classical quantum groups. Also, we show that several structural properties
of $\mathbb{G}$ are encoded by $\tilde \mathbb{G}$: the latter, despite being a
simpler object, can carry very important information about $\mathbb{G}$.
