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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?

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    $\begingroup$ The question seems like a mixture of some concepts that are appropriate to SR but not to GR and other concepts that are appropriate to GR but not to SR. GR doesn't have transformations in which we scale the coordinates, or any other groups of continuous or discrete symmetries that apply to all spacetimes (except for the diffeomorphisms, and that's kind of vacuous). SR doesn't allow nontrivial conformal transformations. You might find it relevant to look at section 5.11 of my GR book, lightandmatter.com/genrel , and the nice paper by Dicke referenced there. $\endgroup$
    – user4552
    Dec 24, 2017 at 23:00

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If we consider the global conformal group $${\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}\tag{A}$$ in a space with signature $(p,q)$, restriction to $x$-independent scale factor $\Omega(x)^2$ would exclude the special conformal transformations, i.e. we are left with the product of the Poincare group and the group of dilations.

See also this related Phys.SE post.

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