  Extensive investigation has recently been conducted into a new class of
antiferromagnetic order known as magnetic octupole order. This order may be
concealed due to its higher-rank multipole, making it difficult to detect and
manipulate it by using conventional methods such as the anomalous Hall effect.
In this paper, we propose the nonlinear magnetoelectric effect (NMEE) as a
technique that can detect and manipulate magnetic octupole order because of
identical symmetry constraints. To demonstrate this point, we first classify
the magnetic point groups in terms of magnetic multipoles to identify magnets,
with the lowest-rank multipoles being magnetic octupoles, and reveal many
candidate systems. We then derive the NMEE tensor by using quantum kinetic
theory and show its relation to quantum geometric properties. Finally, we
calculate the NMEE in two antiferromagnets where the lowest-rank nonvanishing
multipole is the magnetic octupole. Specifically, we focus on a $d$-wave
altermagnet and a pyrochlore lattice with all-in/all-out magnetic order,
revealing an enhancement of the NMEE by Weyl points. Finally, we discuss
experimental realization and show that the NMEE has a sizeable value that can
be detected by the magneto-optical Kerr effect.
