We construct maximal supergravity in four dimensions with local scaling symmetry as deformation of the original Cremmer-Julia theory. The different theories which include the standard gaugings are parametrized by an embedding tensor carrying 56+912 parameters. We determine the form of the possible gauge groups and work out the complete set of field equations. As a result we obtain the most general couplings compatible with N=8 supersymmetry in four dimensions. A particular feature of these theories is the absence of an action and an additional positive contribution to the effective cosmological constant. Moreover, these gaugings are generically dyonic, i.e. involve simultaneously electric and magnetic vector fields.
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