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Sean Eberhard
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Converting my comment into an answer:

The $k$th term of the lower central series modulo the $(k+1)$th term is generated by the left-normed commutators $[x_1, \dots, x_k]$ with $x_1, \dots, x_k \in \{g_1, \dots, g_m\}$, of which there are at most $m^{k-1}(m-1)$ for $k > 1$ (since $x_1 \ne x_2$). Therefore $$\dim G \le m + \sum_{k=2}^C m^{k-1}(m-1) = m^C.$$

Sean Eberhard
  • 9.7k
  • 30
  • 57