The collective anisotropic hydrodynamic behavior of charge hadrons is studied using the multiparticle correlation method. The elliptic anisotropy harmonic from different orders of multiparticle cumulants, $v_{2}\{2k\}$, are measured up to the tenth order ($k = 5$) as functions of the collision centrality in lead-lead collisions at an energy of $\sqrt{s_{\mathrm{NN}}}=~5.02$ TeV. The data were obtained by the CMS experiment at the LHC, with an integrated luminosity of $0.58~{\mathrm{nb}}^{-1}$. A fine splitting $v_{2}\{2\}$ > $v_{2}\{4\} \gtrsim v_{2}\{6\} \gtrsim v_{2}\{8\} \gtrsim v_{2}\{10\}$ is observed. The subtle differences in the higher-order cumulants allow for a precise determination of the underlying hydrodynamics. Based on these results, centrality-dependent moments for the fluctuation-driven event-by-event $v_{2}$ distribution are determined, including the skewness, the kurtosis and, for the first time, the superskewness. Assuming a hydrodynamic expansion of the produced medium, these moments probe the initial-state geometry in high-energy nucleus-nucleus collisions.