  We present a global fit for $\alpha_s(m_Z)$, analyzing the available
C-parameter data measured at center-of-mass energies between $Q=35$ and $207$
GeV. The experimental data is compared to a N$^3$LL$^\prime$ +
$\mathcal{O}(\alpha_s^3)$ + $\Omega_1$ theoretical prediction (up to the
missing 4-loop cusp anomalous dimension), which includes power corrections
coming from a field theoretical nonperturbative soft function. The dominant
hadronic parameter is its first moment $\Omega_1$, which is defined in a scheme
which eliminates the $\mathcal{O}(\Lambda_{\rm QCD})$ renormalon ambiguity. The
resummation region plays a dominant role in the C-parameter spectrum, and in
this region a fit for $\alpha_s(m_Z)$ and $\Omega_1$ is sufficient. We find
$\alpha_s(m_Z)=0.1123\pm 0.0015$ and $\Omega_1=0.421\pm 0.063\,{\rm GeV}$ with
$\chi^2/\rm{dof}=0.988$ for $404$ bins of data. These results agree with the
prediction of universality for $\Omega_1$ between thrust and C-parameter within
1-$\sigma$.
