  The hexadecapole deformation, as well as the quadrupole one, influences the
low-lying states of finite nuclei. The hexadecapole correlations are often
overshadowed by the large quadrupole effects, and hence have not been much
investigated. Here we address the relevance of hexadecapole deformations in the
calculations of low-energy collective states of heavy nuclei, by using the
theoretical framework of the self-consistent mean-field method and the
interacting-boson approximation. The interacting-boson Hamiltonian that
explicitly includes the quadrupole and hexadecapole collective degrees of
freedom is specified by a choice of the energy density functional and pairing
interaction. In an illustrative application to axially deformed Gd isotopes, it
is shown that the inclusion of the hexadecapole degree of freedom does not
affect most of the low-spin and low-lying states qualitatively, but that has
notable effects in that it significantly improves the description of high-spin
states of the ground-state bands of nearly spherical vibrational nuclei and
gives rise to $K^\pi=4^+$ bands exhibiting strong $E4$ transitions in strongly
deformed nuclei.
