We report on the measurement of spin density matrix elements of the $Lambda(1520)$ in the photoproduction reaction $gamma prightarrow Lambda(1520)K^+$, via its subsequent decay to $K^{-}p$. The measurement was performed as part of the GlueX experimental program in Hall D at Jefferson Lab using a linearly polarized photon beam with $E_gamma =$ 8.2-8.8 GeV. These are the first such measurements in this photon energy range. Results are presented in bins of momentum transfer squared, $-(t-t_text{0})$. We compare the results with a Reggeon exchange model and determine that natural exchange amplitudes are dominant in $Lambda(1520)$ photoproduction.