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 (FeO) 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 Fe vacancies. However, due to the high quality of these samples, AC magnetometry measurements have shown excellent specific absorption rates (>1000 W g 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.