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.
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