  We construct the K=8 fractional superconformal algebras. There are two such
extended Virasoro algebras, one of which was constructed earlier, involving a
fractional spin (equivalently, conformal dimension) 6/5 current. The new
algebra involves two additional fractional spin currents with spin 13/5. Both
algebras are non-local and satisfy non-abelian braiding relations. The
construction of the algebras uses the isomorphism between the Z_8 parafermion
theory and the tensor product of two tricritical Ising models. For the special
value of the central charge c=52/55, corresponding to the eighth member of the
unitary minimal series, the 13/5 currents of the new algebra decouple, while
two spin 23/5 currents (level-2 current algebra descendants of the 13/5
currents) emerge. In addition, it is shown that the K=8 algebra involving the
spin 13/5 currents at central charge c=12/5 is the appropriate algebra for the
construction of the K=8 (four-dimensional) fractional superstring.
