  Electronic Fabry-P{\'e}rot interferometry is a powerful method to probe
quasiparticle charge and anyonic braiding statistics in the fractional quantum
Hall regime. We extend this technique to the hierarchy $\nu = 2/5$ fractional
quantum Hall state, possessing two edge modes that in our device can be
interfered independently. The outer edge mode exhibits interference similar to
the behavior observed at the $\nu = 1/3$ state, indicating that the outer edge
mode at $\nu = 2/5$ has properties similar to the single mode at $\nu = 1/3$.
The inner mode shows an oscillation pattern with a series of discrete phase
jumps indicative of distinct anyonic braiding statistics. After taking into
account the impact of bulk-edge coupling, we extract an interfering
quasiparticle charge ${e^*} = 0.17 \pm 0.02$ and anyonic braiding phase $\theta
_a = (-0.43 \pm 0.05)\times 2\pi$, which serve as experimental verification of
the theoretically predicted values of $e^* = \frac{1}{5}$ and $\theta _a =
-\frac{4\pi}{5}$.
