  The high temperature magnetic order in SrRu$_2$O$_6$ was studied by measuring
magnetization and neutron powder diffraction with both polarized and
unpolarized neutrons. SrRu$_2$O$_6$ crystallizes into the hexagonal lead
antimonate (PbSb$_2$O$_6$, space group \textit{P}$\overline{3}$1\textit{m})
structure with layers of edge-sharing RuO$_6$ octahedra separated by Sr$^{2+}$
ions. SrRu$_2$O$_6$ orders at $T_N$=565\,K with Ru moments coupled
antiferromagnetically both in-plane and out-of-plane. The magnetic moment is
1.30(2) $\mu_\mathrm{B}$/Ru at room temperature and is along the
crystallographic \textit{c}-axis in the G-type magnetic structure. We performed
density functional calculations with constrained RPA to obtain the electronic
structure and effective intra- and inter-orbital interaction parameters. The
projected density of states show strong hybridization between Ru 4$d$ and O
2$p$. By downfolding to the target $t_{2g}$ bands we extracted the effective
magnetic Hamiltonian. We performed Monte Carlo simulations to determine the
transition temperature as a function of inter- and intra-plane couplings and
find weak inter plane coupling, 3\% of the intra-plane coupling, permits three
dimensional magnetic order at $T_N$. As suggested by the magnetic
susceptibility, two-dimensional correlations persist above $T_N$ due to the
strong intra-plane coupling.
