= Solution
For <gamma decay>, the <nuclear gamma-ray multipole selection rules> require the radiation's <angular momentum> $L$ to satisfy the triangle inequality
$$
|J_i-J_f|\leq L\leq J_i+J_f.
$$
Here $J_i=J_f=5/2$, so $0\leq L\leq5$. A real emitted <photon> has no monopole radiation mode, leaving $L=1,2,3,4,5$.
The two nuclear <quantum states> have the same <parity>. An electric multipole has parity factor $(-1)^L$, while a magnetic multipole has factor $(-1)^{L+1}$. Thus electric radiation requires even $L$ and magnetic radiation odd $L$:
$$
\boxed{M1,\quad E2,\quad M3,\quad E4,\quad M5.}
$$
In particular, an $E0$ nuclear transition cannot emit the single real <photon> requested here; processes such as <internal conversion> are different decay channels.
For usual nuclear <gamma ray> energies, $kR\ll1$, where $k=E_\gamma/(\hbar c)$ and $R$ is the nuclear size. Higher <multipole radiation> orders bring extra powers of this small parameter, so \b[$M1$ and $E2$ are the likely important contributions], with $M3$, $E4$ and $M5$ generally much smaller. Often $M1$ is dominant, but an allowed $M1$ <matrix element> can be hindered and an $E2$ transition can be collectively enhanced. The <quantum numbers> alone do not fix a numerical relative contribution or an unconditional ordering of the $M1$ and $E2$ rates.
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