We study rheology, microstructure, and response to an applied electric field (E) in suspensions of fumed alumina (Al2O3) nanoparticles in a nematic liquid crystal (NLC) made of N-(4-methoxybenzylidene)-4-butylaniline (MBBA). Fumed Al2O3/MBBA suspensions exhibit flowability with nanoparticle volume fraction (phi)=0.001 and 0.007, and become solid-like gels at a nanoparticle phi=0.014 and beyond. The dynamic rheology of gel-like suspensions follows the soft glass rheology (SGR) model. The effective noise temperature remains close to 1 for these Al2O3/MBBA suspensions, which serves as an indication of the presence of glassy dynamics. Further, the optical microscopy and the differential scanning calorimetry (DSC) reveal that the incorporation of fumed Al2O3 nanoparticles causes a significant depression in the nematic-isotropic phase transition temperature (T-NI). The gel-like suspensions are less sensitive to pre-shear and show a large structural recovery after shear when compared to Al2O3/silicone oil suspensions. At and above a critical nanoparticle phi=0.005, the suspensions exhibit a reversible nematic-isotropic phase transition under the application of E.