  Although $SO(10)$ Supersymmetric (SUSY) Grand Unification Theories (GUTs) are
very attractive for neutrino mass and mixing, it is often quite difficult to
achieve successful leptogenesis from the lightest right-handed neutrino $N_1$
due to the strong relations between neutrino and up-type quark Yukawa
couplings. We show that in a realistic model these constraints are relaxed,
making $N_1$ leptogenesis viable. To illustrate this, we calculate the baryon
asymmetry of the Universe $ Y_B $ from flavoured $ N_1 $ leptogenesis in a
recently proposed $ \Delta(27) \times SO(10) $ SUSY GUT. The flavoured
Boltzmann equations are solved numerically, and comparison with the observed $
Y_B $ places constraints on the allowed values of right-handed neutrino masses
and neutrino Yukawa couplings. The flavoured $SO(10)$ SUSY GUT is not only
fairly complete and predictive in the lepton sector, but can also explain the
BAU through leptogenesis with natural values in the lepton sector albeit with
some tuning in the quark sector.
