extension of the <span style="font-variant:small-caps;">Powheg V2</span> generator based on the <span style="font-variant:small-caps;">MiNLO</span> prescription [@Hamilton:2012np] for the improved next-to-leading-logarithmic accuracy applied to the ${\ensuremath{\Pg \Pg \to \PH}\xspace}$ production in association with up to one additional jet. It provides a description of the ${\ensuremath{\Pg \Pg \to \PH}\xspace}$ production in association with zero jets and one jet at NLO accuracy, and the ${\ensuremath{\Pg \Pg \to \PH}\xspace}$ production in association with two jets only at the leading-order (LO) accuracy. All the generators used to describe the ${\ensuremath{\Pg \Pg \to \PH}\xspace}$ process take into account the finite masses of the bottom and top quarks. The description of the SM Higgs boson production in the vector boson fusion (VBF) process is obtained at NLO accuracy using the generator. The processes of SM Higgs boson production associated with gauge bosons ([$\mathrm{V} \PH$]{}) or top quark-antiquark pair (${\ensuremath{\PQt \PAQt \PH}\xspace}$) are described at LO accuracy using <span style="font-variant:small-caps;">Pythia 6.4</span> [@Sjostrand:2006za]. The MC samples simulated with these generators are normalized to the inclusive SM Higgs boson production cross sections and branching fractions that correspond to the SM calculations at NNLO and NNLL accuracy, in accordance with the LHC Higgs Cross Section Working Group recommendations [@Heinemeyer:2013tqa].The samples of the ${\ensuremath{\Pg 