  We consider in any dimension the supersymmetric $\mathbb{Z}_2$ truncations of
the maximal supergravity theories. In each dimension and for each truncation we
determine all the sets of 1/2-BPS space-filling branes that preserve the
supersymmetry of the truncated theory and the representations of the symmetry
of such theory to which they belong. We show that in any dimension below eight
these sets always contain exotic branes, that are objects that do not have a
ten-dimensional origin. We repeat the same analysis for half-maximal theories
and for the quarter-maximal theories in four and three dimensions. We then
discuss all the possible gaugings of these theories as described in terms of
the embedding tensor. In general, the truncation acts on the quadratic
constraints of the embedding tensor in such a way that some representations
survive the truncation although they are not required by the supersymmetry of
the truncated theory. We show that for any theory, among these representations,
the highest-dimensional ones are precisely those of the 1/2-BPS space-filling
branes that preserve the same supersymmetry of the truncated theory, and we
interpret this result as the fact that these quadratic constraints after the
truncation become tadpole conditions for such branes.
