The development of new approaches to tuning the resonant magnetic response of simple all-dielectric nanostructures is very important in modern nanophotonics. Here, it is shown that a resonant magnetic octupole (MOCT) response can be obtained by dividing a solid rectangular silicon block to a quadrumer structure with the introduction of narrow gaps between four nanocubes. The spectral position of the MOCT resonance is controlled and tuned by varying the distance between the nanocubes. It is demonstrated that several magnetic hot-spots related to the MOCT resonance can be located in the gaps creating a strong magnetic field gradient in free space. It is observed that the resonant excitation of the MOCT moment leads to a significant enhancement of light absorption in the system at the spectral region, where light absorption in bulk silicon is weak. The results of this work can be applied to design new composite antennas and metamaterials based on complex building blocks, energy harvesting devices, and molecular trapping with magnetic hot-spots.