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Mott insulators are paradigms of strongly correlated physics, giving rise to phases of matter with novel and hard-to-explain properties. Extending the typical SU(2) symmetry of Mott insulators to SU($N$) is predicted to give exotic quantum magnetism at low temperatures, but understanding the effect of strong quantum fluctuations for large $N$ remains an open challenge. In this work, we experimentally observe nearest-neighbor spin correlations in the SU(6) Hubbard model realized by ytterbium atoms in optical lattices. We study one-dimensional, two-dimensional square, and three-dimensional cubic lattice geometries. The measured SU(6) spin correlations are dramatically enhanced compared to the SU(2) correlations, due to strong Pomeranchuk cooling. We also present numerical calculations based on exact diagonalization and determinantal quantum Monte Carlo. The experimental data for a one-dimensional lattice agree with theory, without any fitting parameters. The detailed comparison between theory and experiment allows us to infer from the measured correlations a lowest temperature of $\left[{0.096 \pm 0.054 \, \rm{(theory)} \pm 0.030 \, \rm{(experiment)}}\right]/k_{\rm B}$ times the tunneling amplitude. For two- and three-dimensional lattices, experiments reach entropies below where our calculations converge, highlighting the experiments as quantum simulations. These results open the door for the study of long-sought SU($N$) quantum magnetism.