  We use lattice simulations and the continuous renormalization-group method,
based on the gradient flow, to calculate the $\beta$ function and anomalous
dimensions of the SU(3) gauge theory with $N_f=10$ flavors of fermions in the
fundamental representation. We employ several improvements to extend the range
of available renormalized couplings, including the addition of heavy
Pauli-Villars bosons to reduce cutoff effects and the combination of a range of
gradient flow transformations. While in the weak coupling regime our result is
consistent with those of earlier studies, our techniques allow us to study the
system at much stronger couplings than previously possible. We find that the
renormalization group $\beta$ function develops a zero, corresponding to an
infrared-stable fixed point, at gradient-flow coupling $g^2=15.0(5)$. We also
determine the mass and tensor anomalous dimensions: At the fixed point we find
$\gamma_m\simeq0.6$, suggesting that this system might be deep inside the
conformal window.
