  Background: The influence of halo structure of $^6$He, $^8$B, $^{11}$Be and
$^{11}$Li nuclei in several mechanisms such as direct reactions and fusion is
already established, although not completely understood. The influence of the
$^{10}$C Brunnian structure is less known.
  Purpose: To investigate the influence of the cluster configuration of
$^{10}$C on the elastic scattering at an energy close to the Coulomb barrier.
  Methods: We present experimental data for the elastic scattering of the
$^{10}$C+$^{208}$Pb system at $E_{\rm lab}$ = 66 MeV. The data are compared to
the three- and the four-body continuum-discretized coupled-channels
calculations assuming $^9$B+$p$, $^6$Be+$\alpha$ and $^8$Be+$p$+$p$
configurations.
  Results: The experimental angular distribution of the cross sections shows
the suppression of the Fresnel peak that is reasonably well reproduced by the
continuum-discretized coupled-channels calculations. However, the calculations
underestimate the cross sections at backward angles. Couplings to continuum
states represent a small effect.
  Conclusions: The cluster configurations of $^{10}$C assumed in the present
work are able to describe some of the features of the data. In order to explain
the data at backward angles, experimental data for the breakup and an extension
of theoretical formalism towards a four-body cluster seem to be in need to
reproduce the measured angular distribution.
