  The complete structure of the $WG_2$ algebra is obtained from an explicit
realization by an abstract Virasoro algebra and a free boson field. We then
construct its BRST operator and find a seven-parameter family of nilpotnt BRST
operators. These free parameters are related to the canonical transformations
of the ghost, antighost fields which leave the total stress-energy tensor and
the antighost field $b$ invariant.
