  We present new HST optical and near-infrared (NIR) photometry of the rich
globular cluster (GC) system of NGC 4874, the cD galaxy in the core of the Coma
cluster (Abell 1656). NGC 4874 was observed with the HST Advanced Camera for
Surveys in the F475W (g) and F814W (I) passbands and the Wide Field Camera 3 IR
Channel in F160W (H). The GCs in this field exhibit a bimodal optical color
distribution with more than half of the GCs falling on the red side at g-I > 1.
Bimodality is also present, though less conspicuously, in the optical-NIR I-H
color. Consistent with past work, we find evidence for nonlinearity in the g-I
versus I-H color-color relation. Our results thus underscore the need for
understanding the detailed form of the color-metallicity relations in
interpreting observational data on GC bimodality. We also find a very strong
color-magnitude trend, or "blue tilt," for the blue component of the optical
color distribution of the NGC 4874 GC system. A similarly strong trend is
present for the overall mean I-H color as a function of magnitude; for M_814 <
-10 mag, these trends imply a steep mass-metallicity scaling with $Z\propto
M_{\rm GC}^{1.4\pm0.4}$, but the scaling is not a simple power law and becomes
much weaker at lower masses. As in other similar systems, the spatial
distribution of the blue GCs is more extended than that of the red GCs, partly
because of blue GCs associated with surrounding cluster galaxies. In addition,
the center of the GC system is displaced by 4+/-1 kpc towards the southwest
from the luminosity center of NGC 4874, in the direction of NGC 4872. Finally,
we remark on a dwarf elliptical galaxy with a noticeably asymmetrical GC
distribution. Interestingly, this dwarf has a velocity of nearly -3000 km/s
with respect to NGC 4874; we suggest it is on its first infall into the cluster
core and is undergoing stripping of its GC system by the cluster potential.
