  We report on the measurement of the isotope shifts of the $3d^74s \,\, a \,
{}^5\!F_5 - 3d^74p \,\, z \, {}^5\!G^o_6$ Fe~I line at 358~nm between all four
stable isotopes ${}^{54}$Fe, ${}^{56}$Fe, ${}^{57}$Fe and ${}^{58}$Fe, as well
as the hyperfine structure of that line for ${}^{57}$Fe, the only stable
isotope having a nonzero nuclear spin. This line is of primary importance for
laser cooling applications. In addition, an experimental value of the field and
specific mass shift coefficients of the transition is reported as well as the
hyperfine structure magnetic dipole coupling constant $A$ of the transition
excited state in $^{57}$Fe, namely $A(3d^74p \,\, z \, {}^5\!G^o_6)=31.241(48)$
MHz. The measurements were carried out by means of laser-induced fluorescence
spectroscopy performed on an isotope-enriched iron atomic beam. All measured
frequency shifts are reported with uncertainties below the third percent level.
