  Centaurus B is a nearby radio galaxy positioned in the Southern hemisphere
close to the Galactic plane. Here we present a detailed analysis of about 43
months of accumulated Fermi-LAT data of the gamma-ray counterpart of the source
initially reported in the 2nd Fermi-LAT catalog, and of newly acquired Suzaku
X-ray data. We confirm its detection at GeV photon energies, and analyze the
extension and variability of the gamma-ray source in the LAT dataset, in which
it appears as a steady gamma-ray emitter. The X-ray core of Centaurus B is
detected as a bright source of a continuum radiation. We do not detect however
any diffuse X-ray emission from the known radio lobes, with the provided upper
limit only marginally consistent with the previously claimed ASCA flux. Two
scenarios that connect the X-ray and gamma-ray properties are considered. In
the first one, we assume that the diffuse non-thermal X-ray emission component
is not significantly below the derived Suzaku upper limit. In this case,
modeling the inverse-Compton emission shows that the observed gamma-ray flux of
the source may in principle be produced within the lobes. This association
would imply that efficient in-situ acceleration of the radiating electrons is
occurring and that the lobes are dominated by the pressure from the
relativistic particles. In the second scenario, with the diffuse X-ray emission
well below the Suzaku upper limits, the lobes in the system are instead
dominated by the magnetic pressure. In this case, the observed gamma-ray flux
is not likely to be produced within the lobes, but instead within the nuclear
parts of the jet. By means of synchrotron self-Compton modeling we show that
this possibility could be consistent with the broad-band data collected for the
unresolved core of Centaurus B, including the newly derived Suzaku spectrum.
