  Minimal unitary representation of SO*(8) = SO(6,2) realized over the Hilbert
space of functions of five variables and its deformations labeled by the spin t
of an SU(2) subgroup correspond to massless conformal fields in six dimensions
as was shown in arXiv:1005.3580. In this paper we study the minimal unitary
supermultiplet of OSp(8*|2N) with the even subgroup SO*(8) x USp(2N) and its
deformations using quasiconformal methods. We show that the minimal unitary
supermultiplet of OSp(8*|2N) admits deformations labeled uniquely by the spin t
of an SU(2) subgroup of the little group SO(4) of lightlike vectors in six
dimensions. We construct the deformed minimal unitary representations and show
that they correspond to massless 6D conformal supermultiplets. The minimal
unitary supermultiplet of OSp(8*|4) is the massless supermultiplet of (2,0)
conformal field theory that is believed to be dual to M-theory on AdS_7 x S^4.
We study its deformations in further detail and show that they are isomorphic
to the doubleton supermultiplets constructed by using twistorial oscillators.
