  In 1980 Rostislav Grigorchuk constructed a group $G$ of intermediate growth,
and later obtained the following estimates on its growth $\gamma$:
$e^{\sqrt{n}}\precsim\gamma(n)\precsim e^{n^\beta},$ where
$\beta=\log_{32}(31)\approx0.991$. He conjectured that the lower bound is
actually tight. In this paper we improve the lower bound to
$e^{n^\alpha}\precsim\gamma(n),$ where $\alpha\approx0.5157$, with the aid of a
computer. This disproves the conjecture that the lower bound be tight.
