  We study numerically the fractal dimensions and the bulk three-point
connectivity for the spin clusters of the Q-state Potts model in two dimensions
with $1\leq Q\leq 4$. We check that the usually invoked correspondence between
FK clusters and spin clusters works at the level of fractal dimensions.
However, the fine structure of the conformal field theories describing critical
clusters first manifests at the level of the three-point connectivities.
Contrary to what recently found for FK clusters, no obvious relation emerges
for generic Q between the spin cluster connectivity and the structure constants
obtained from analytic continuation of the minimal model ones. The numerical
results strongly suggest then that spin and FK clusters are described by
conformal field theories with different realizations of the color symmetry of
the Potts model.
