Let $ G $ be a quasisimple finite group. Let $ d_{min} $ be the minimum dimension of a nontrivial irrep of $ G $. Must it be the case that the image of all (nontrivial) dimension $ d_{min} $ irreps of $ G $ are conjugate in $ SU(d_{min}) $?
For example all $ SL(2,5) $ subgroups of $ SU(2) $ are conjugate. This is especially interesting given that there are actually two distinct irreps of $ SL(2,5) $ into $ SU(2) $, they just happen to have conjugate image.
And I'm pretty sure that all $ A_5 $ subgroups of $ SU(3) $ are conjugate and that all $ PSL(2,7) $ subgroups of $ SU(3) $ are conjugate and that all $ 3.A_6 $ subgroups of $ SU(3) $ are conjugate.
And I'm pretty sure that all $ SL(2,9) \cong 2.A_6 $ and $ 2.A_7 $ and $ Sp(4,3) $ subgroups of $ SU(4) $ are also unique up to conjugacy
The irrep of $ PSL(2,8) $ with character $ (7,−1,−2,1,1,1,0,0,0) $ has an image which is not conjugate to the images of the three other degree $ 7 $ irreps.
Since the representation is irreducible, Schur's lemma shows that the only way the images can be conjugate is for there to be an automorphism that takes one character to the other. But the outer automorphism group of $ PSL(2,8) $ is size $ 3 $. So the four degree $ 7 $ characters cannot all have conjugate image.
This answer is from Dave Benson given where I cross posted my question to MO https://mathoverflow.net/a/445933/387190