  We use the ROSAT All-Sky Survey to study the X-ray properties of a sample of
625 groups and clusters of galaxies selected from the Sloan Digital Sky Survey.
We stack clusters with similar velocity dispersions and investigate whether
their average X-ray luminosities and surface brightness profiles vary with the
radio activity level of their central galaxies. We find that at a given value
of $\sigma$, clusters with a central radio AGN have more concentrated X-ray
surface brightness profiles, larger central galaxy masses, and higher X-ray
luminosities than clusters with radio-quiet central galaxies. The enhancement
in X-ray luminosity is more than a factor of two, is detected with better than
6$\sigma$ significance, and cannot be explained by X-ray emission from the
radio AGN itself. This difference is largely due to a subpopulation of
radio-quiet, high velocity dispersion clusters with low mass central galaxies.
These clusters are underluminous at X-ray wavelengths when compared to
otherwise similar clusters where the central galaxy is radio-loud, more
massive, or both.
