  Measuring the hyperfine structure (HFS) of long-lived $^3P_2$ states of
divalent atoms may offer the opportunity of extracting relatively unexplored
nuclear magnetic octupole and electric hexadecapole moments. Here, using
relativistic many-body methods of atomic structure and the nuclear shell model,
we evaluate the effect of these higher nuclear moments on the hyperfine
structure. We find that the sensitivity of HFS interval measurements in
$^{87}$Sr needed to reveal the perturbation caused by the nuclear octupole
moment is on the order of kHz. Results of similar analyses for $^{9}$Be,
$^{25}$Mg, and $^{43}$Ca are also reported.
