Most studies on magnetic nanoparticle-based hyperthermia utilize iron oxide nanoparticles smaller than 20 nm, which are intended to have superparamagnetic behavior (SP-MNPs). However, the heating power of larger magnetic nanoparticles with non-fluctuating or fixed magnetic dipoles (F-MNPs) can be significantly greater than that of SP-MNPs if high enough fields (H > 15 mT) are used. But the synthesis of larger single nanocrystals of magnetite (Fe3O4) with a regular shape and narrow size distribution devoid of secondary phases remains a challenge. Iron oxide nanoparticles, grown over 25 nm, often present large shape and size polydispersities, twinning defects and a significant fraction of the wüstite-type (FeO) paramagnetic phase, resulting in degradation of magnetic properties. Herein, we introduce an improved procedure to synthesize monodisperse F-MNPs in the range of 25 to 50 nm with a distinct octahedral morphology and very crystalline magnetite phase. We unravel the subtle phase transformation that takes place during the synthesis by a thorough study in several non-optimized nanoparticles presenting a core-shell structure or composed of magnetite-type clusters embedded in a wüstite lattice. Optimized magnetite samples present a slight decrease in the saturation magnetization compared to bulk magnetite, which is successfully explained by the presence of Fe2+ vacancies. However, due to the high quality of these samples, AC magnetometry measurements have shown excellent specific absorption rates (>1000 W gFe3O4-1 at 40 mT and 300 kHz). Most importantly, the magnetic response and the hyperthermia performance of properly coated F-MNPs are kept basically unaltered in media with very different viscosities and ionic strength. Finally, using a physical model based on single magnetic domain approaches, we derive a novel connection between the octahedral shape and the high hyperthermia performance.