No. See e.g.
https://arxiv.org/abs/math-ph/0102011
In this paper the authors perform a finite general coordinate transformation (i.e. not an infinitesimal one to derive the algebra, but a finite to derive the group at once) on the action of a point particle, and solve the resulting differential equation stating that the Lagrangian should be kept invariant up to a total time derivative. It's been a while for me (so correct me if I'm wrong), but I'd say this group is the Schrodinger group.
To give a simple counter example: take the action of the point particle, and rescale the time as
t \rightarrow t'= z^{\alpha} t, \ \ \ \ x^i \ \rightarrow x^{'i} = z^{\beta} x^i.
and then solve for the coefficients ##\alpha## and ##\beta## by the demand that the action stays invariant. This transformation, called a dilation, is not in the Galilei-group. Another transformation would be the one given by a rescaling of the mass. This transformation is not described by a spacetime transformation, but by the action of the so-called central extension of the Galilei-algebra called the Bargmann algebra.