  The properties of a nanosecond isomer in $^{32}$Si, disputed in previous
studies, depend on the evolution of proton and neutron shell gaps near the
`island of inversion'. We have placed the isomer at 5505.2(2) keV with $J^{\pi}
= 5^-$, decaying primarily via an $E3$ transition to the $2^+_1$ state. The
$E3$ strength of 0.0841(10) W.u. is unusually small and suggests that this
isomer is dominated by the $(\nu d_{3/2})^{-1} \otimes (\nu f_{7/2})^{1}$
configuration, which is sensitive to the $N=20$ shell gap. A newly observed
$4^+_1$ state is placed at 5881.4(13) keV; its energy is enhanced by the $Z=14$
subshell closure. This indicates that the isomer is located in a `yrast trap',
a feature rarely seen at low mass numbers.
