  We present a study of the connection between brightest cluster galaxies
(BCGs) and their host galaxy clusters. Using galaxy clusters at $0.1<z<0.3$
from the Hectospec Cluster Survey (HeCS) with X-ray information from the
Archive of {\it Chandra} Cluster Entropy Profile Tables (ACCEPT), we confirm
that BCGs in low central entropy clusters are well aligned with the X-ray
center. Additionally, the magnitude difference between BCG and the 2nd
brightest one also correlates with the central entropy of the intracluster
medium. From the red-sequence (RS) galaxies, we cannot find significant
dependence of RS color scatter and stellar population on the central entropy of
the intracluster medium of their host cluster. However, BCGs in low entropy
clusters are systematically less massive than those in high entropy clusters,
although this is dependent on the method used to derive the stellar mass of
BCGs. In contrast, the stellar velocity dispersion of BCGs shows no dependence
on BCG activity and cluster central entropy. This implies that the potential of
the BCG is established earlier and the activity leading to optical emission
lines is dictated by the properties of the intracluster medium in the cluster
core.
