The group $SO(d,2)$ preserves the Minkowski space $\mathbb{R}^{d-1,1}$ up to a function $\Omega(x)^2$, that depends on the position co-ordinates.
$$ds^2 \rightarrow \Omega(x)^2 ds^2.$$
What group preserves the metric just up to a constant factor say $\lambda$?
$$ds^2 \rightarrow \lambda^2 ds^2~?$$
I can see that the usual Lorentz and scale transformations will do. But, are there any other non-trivial transformations (like special conformal for the conformal group)? And what group are we talking about?