  Using a sample of 123 X-ray clusters and groups drawn from the XMM-Cluster
Survey first data release, we investigate the interplay between the brightest
cluster galaxy (BCG), its black hole, and the intra-cluster/group medium (ICM).
It appears that for groups and clusters with a BCG likely to host significant
AGN feedback, gas cooling dominates in those with Tx > 2 keV while AGN feedback
dominates below. This may be understood through the sub-unity exponent found in
the scaling relation we derive between the BCG mass and cluster mass over the
halo mass range 10^13 < M500 < 10^15Msol and the lack of correlation between
radio luminosity and cluster mass, such that BCG AGN in groups can have
relatively more energetic influence on the ICM. The Lx - Tx relation for
systems with the most massive BCGs, or those with BCGs co-located with the peak
of the ICM emission, is steeper than that for those with the least massive and
most offset, which instead follows self-similarity. This is evidence that a
combination of central gas cooling and powerful, well fuelled AGN causes the
departure of the ICM from pure gravitational heating, with the steepened
relation crossing self-similarity at Tx = 2 keV. Importantly, regardless of
their black hole mass, BCGs are more likely to host radio-loud AGN if they are
in a massive cluster (Tx > 2 keV) and again co-located with an effective fuel
supply of dense, cooling gas. This demonstrates that the most massive black
holes appear to know more about their host cluster than they do about their
host galaxy. The results lead us to propose a physically motivated, empirical
definition of 'cluster' and 'group', delineated at 2 keV.
