  We present strong numerical evidence for the existence of an infrared fixed
point in the renormalization group flow of the SU(3) gauge-fermion system with
twelve massless fermions in the fundamental representation. Our numerical
simulations using nHYP-smeared staggered fermions with Pauli-Villars
improvement do not exhibit any first-order bulk phase transition in the
investigated parameter region. We utilize an infinite volume renormalization
scheme based on the gradient flow transformation to determine the
renormalization group $\beta$ function. We identify an infrared fixed point at
$g^2_{\mathrm{GF}\star}=6.60(62)$ in the GF scheme and calculate the leading
irrelevant critical exponent $\gamma_{g}^{\star}=0.199(32)$. Our prediction for
$\gamma_{g}^{\star}$ is consistent with available literature at the
$1\mbox{-}2\sigma$ level.
