  Masses of $^{52g,52m}$Co were measured for the first time with an accuracy of
$\sim 10$ keV, an unprecedented precision reached for short-lived nuclei in the
isochronous mass spectrometry. Combining our results with the previous
$\beta$-$\gamma$ measurements of $^{52}$Ni, the $T=2$, $J^{\pi}=0^+$ isobaric
analog state (IAS) in $^{52}$Co was newly assigned, questioning the
conventional identification of IASs from the $\beta$-delayed proton emissions.
Using our energy of the IAS in $^{52}$Co, the masses of the $T=2$ multiplet fit
well into the Isobaric Multiplet Mass Equation. We find that the IAS in
$^{52}$Co decays predominantly via $\gamma$ transitions while the proton
emission is negligibly small. According to our large-scale shell model
calculations, this phenomenon has been interpreted to be due to very low
isospin mixing in the IAS.
