  We present a novel global E_7(7) symmetry in five-dimensional maximal
supergravity as well as an E_8(8) symmetry in d=4. These symmetry groups which
are known to be present after reduction to d=4 and d=3, respectively, appear as
conformal extensions of the respective well-known hidden-symmetry groups. A
global scaling symmetry of the Lagrangian is the key to enhancement of E_6(6)
to E_7(7) in d=5 and E_7(7) to E_8(8) in d=4. The group action on the physical
fields is induced by conformal transformations in auxiliary spaces of
dimensions 27 and 56, respectively. The construction is analogous to the one
where the conformal group of Minkowski space acts on the boundary of AdS_5
space. A geometrical picture underlying the action of these ``conformal duality
groups'' is given.
