The outburst of Nova Sgr 2011 N.2 (=V5588 Sgr) was followed with optical and near-IR photometric and spectroscopic observations for 3.5 yr, beginning shortly before the maximum. V5588 Sgr is located close to Galactic Centre, suffering from E(B-V)=1.56(+/-0.1) extinction. The primary maximum was reached at V=12.37 on UT 2011 April 2.5(+/-0.2), and the underlying smooth decline was moderately fast with t^V^_2_=38 and t^V^_3_=77 d. On top of an otherwise normal decline, six self-similar, fast evolving and bright secondary maxima (SdM) appeared in succession. Only very few other novae have presented so clear SdM. Both the primary maximum and all SdM occurred at later times with increasing wavelengths, by amounts in agreement with expectations from fireball expansions. The radiative energy released during SdM declined following an exponential pattern, while the breadth of individual SdM and the time interval between them widened. Emission lines remained sharp (FWHM~1000 km/s) throughout the whole nova evolution, with the exception of a broad pedestal with a trapezoidal shape ({Delta}vel=3600 km/s at the top and 4500 km/s at the bottom) which was only seen during the advanced decline from SdM maxima and was absent in between SdM. V5588 Sgr at maximum light displayed a typical FeII-class spectrum which did not evolve into a nebular stage. About 10 d into the decline from primary maximum, a typical high-ionization He/N-class spectrum appeared and remained visible simultaneously with the FeII-class spectrum, qualifying V5588 Sgr as a rare hybrid nova. While the FeII-class spectrum faded into oblivion, the He/N-class spectrum developed strong [FeX] coronal lines.