Magnetic octupole Hall effect in heavy transition metals

$d$-wave altermagnets have the magnetic octupole as their primary order parameter. A recent study [Han et al., Phys. Rev. Lett. 135, 076705 (2025)] demonstrated that magnetic octupole current can induce N\'eel vector dynamics. Therefore, identifying materials that can efficiently generate a magnetic octupole current is essential. In this paper, we investigate the magnetic octupole Hall effect in $4d$ and $5d$ transition metals. By employing atomic magnetic octupole operators, we calculate the magnetic octupole Hall conductivity using first-principles calculations. We also explore the microscopic origin of the magnetic octupole Hall effect and find that it results from the combined effect of orbital texture and spin-orbit coupling. Additionally, we compare the spin Hall conductivity with magnetic octupole Hall conductivity across various materials and identify those that are optimal for observing magnetic octupole physics. We also discuss potential applications arising from the magnetic octupole Hall effect. Our work serves as a valuable reference for identifying materials suitable for studying magnetic octupole physics.