  We use a very large simulation of structure growth in a LCDM universe -- the
Millennium Simulation -- to study assembly bias, the fact that the large-scale
clustering of haloes of given mass varies significantly with their assembly
history. We extend earlier work based on the same simulation by superposing
results for redshifts from 0 to 3, by defining a less noisy estimator of
clustering amplitude, and by considering halo concentration, substructure mass
fraction and spin, as well as formation time, as additional parameters. These
improvements lead to results with less noise than previous studies and covering
a wider range of halo masses and structural properties. We find significant and
significantly different assembly bias effects for all the halo properties we
consider, although in all cases the dependences on halo mass and on redshift
are adequately described as a dependence on equivalent peak height nu(M,z). The
nu-dependences for different halo properties differ qualitatively and are not
related as might naively be expected given the relations between formation
time, concentration, substructure fraction and spin found for the halo
population as a whole. These results suggest that it will be difficult to build
models for the galaxy populations of dark haloes which can robustly relate the
amplitude of large-scale galaxy clustering to that for mass clustering at
better than the 10% level.
