  Abell 407 (A407) is a unique galaxy cluster hosting a central compact group
of nine galaxies (named as 'Zwicky's Nonet'; G1 - G9 in this work) within a 30
kpc radius region. The cluster core also hosts a luminous radio active galactic
nucleus (AGN), 4C 35.06 with helically twisted jets extending over 200 kpc.
With a 44 ks Chandra observation of A407, we characterize the X-ray properties
of its intracluster medium (ICM) and central galaxies. The mean X-ray
temperature of A407 is 2.7 keV and the $M_{200}$ is $1.9 \times 10^{14}
{M_{\odot}}$. We suggest that A407 has a weak cool core at $r < 60$ kpc scales
and at its very center, $< 1$-2 kpc radius, a small galaxy corona associated
with the strong radio AGN. We also conclude that the AGN 4C 35.06 host galaxy
is most likely G3. We suggest that the central group of galaxies is undergoing
a `slow merge' procedure. The range of the merging time-scale is $0.3\sim2.3$
Gyr and the stellar mass of the future brightest cluster galaxy (BCG) will be
$7.4\times10^{11} M_{\odot}$. We find that the regions which overlap with the
radio jets have higher temperature and metallicity. This is consistent with AGN
feedback activity. The central entropy is higher than that for other clusters,
which may be due to the AGN feedback and/or merging activity. With all these
facts, we suggest that A407 is a unique and rare system in the local universe
that could help us to understand the formation of a massive BCG.
