We compute the real-space cluster-galaxy cross-correlation xi_cg(r) using the ROSAT-ESO Flux Limited X-ray (REFLEX) cluster survey, a group catalogue constructed from the final version of the 2dFGRS, and galaxies extracted from 2MASS and APM surveys. This first detailed calculation of the cross-correlation for X-ray clusters and groups, is consistent with previous works and shows that xi_cg(r) can not be described by a single power law. We analyse the clustering dependence on the cluster X-ray luminosity L_X and virial mass M_vir thresholds as well as on the galaxy limiting magnitude. We also make a comparison of our results with those obtained for the halo-mass cross-correlation function in a LambdaCDM N-body simulation to infer the scale dependence of galaxy bias around clusters. Our results indicate that the distribution of galaxies shows a significant anti-bias at highly non-linear small cluster-centric distances (b_cg(r) ~ 0.7), irrespective of the group/cluster virial mass or X-ray luminosity and galaxy characteristics, which show that a generic process controls the efficiency of galaxy formation and evolution in high density regions. On larger scales b_cg(r) rises to a nearly constant value of order unity, the transition occuring at approximately 2 Mpch^(-1) for 2dF groups and 5 Mpch^(-1) for REFLEX clusters.
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