  The generalized supersymmetries admitting abelian bosonic tensorial central
charges are classified in accordance with their division algebra structure
(over ${\bf R}$, ${\bf C}$, ${\bf H}$ or ${\bf O}$). It is shown in particular
that in D=11 dimensions, the $M$-superalgebra admits a consistent octonionic
formulation, involving 52 real bosonic generators (in place of the 528 of the
standard $M$-superalgebra). The octonionic $M5$ (super-5-brane) sector
coincides with the octonionic $M1$ and $M2$ sectors, while in the standard
formulation these sectors are all independent. The octonionic conformal and
superconformal $M$-algebras are explicitly constructed. They are respectively
given by the $Sp(8|{\bf O})$ ($OSp(1,8|{\bf O})$) (super)algebra of
octonionic-valued (super)matrices, whose bosonic subalgebra consists of 232
(and respectively 239) generators.
