  The Gamow-Teller strength distribution of the decay of $^{186}$Hg into
$^{186}$Au has been determined for the first time using the total absorption
gamma spectroscopy technique and has been compared with theoretical QRPA
calculations using the SLy4 Skyrme force. The measured Gamow-Teller strength
distribution and the half-life are described by mixing oblate and prolate
configurations independently in the parent and daughter nuclei. The best
description of the experimental beta strength is obtained with dominantly
prolate components for both parent $^{186}$Hg and daughter $^{186}$Au. The
approach also allowed us to determine an upper limit of the oblate component in
the parent state. The complexity of the analysis required the development of a
new approach in the analysis of the X-ray gated total absorption spectrum.
