  We present a measure of the power spectrum on scales from 15 to 800 Mpc/h
using the ROSAT-ESO Flux-Limited X-Ray(REFLEX) galaxy cluster catalogue. The
REFLEX survey provides a sample of the 452 X-ray brightest southern clusters of
galaxies with the nominal flux limit S=3.0 10^{-12}erg/s/cm2 for the ROSAT
energy band (0.1-2.4)keV. Several tests are performed showing no significant
incompletenesses of the REFLEX clusters with X-ray luminosities brighter than
10^{43}erg/s up to scales of about 800 Mpc/h. They also indicate that cosmic
variance might be more important than previous studies suggest. We regard this
as a warning not to draw general cosmological conclusions from cluster samples
with a size smaller than REFLEX. Power spectra, P(k), of comoving cluster
number densities are estimated for flux- and volume-limited subsamples. The
most important result is the detection of a broad maximum within the comoving
wavenumber range 0.022<k<0.030 h/Mpc. The data suggest an increase of the power
spectral amplitude with X-ray luminosity. Compared to optically selected
cluster samples the REFLEX P(k)is flatter for wavenumbers k<0.05 h/Mpc thus
shifting the maximum of P(k) to larger scales. The smooth maximum is not
consistent with the narrow peak detected at k=0.05 h/Mpc using the Abell/ACO
richness $\ge 0$ data. In the range 0.02<k<0.4 h/Mpc general agreement is found
between the slope of the REFLEX P(k) and those obtained with optically selected
galaxies. A semi-analytic description of the biased nonlinear power spectrum in
redshift space gives the best agreement for low-density Cold Dark Matter models
with or without a cosmological constant.
