I belived it was an 350,
But is it a 500cu
The J and J2 were both 500 = 84 x 90 - (499 actual) cc/cu models.
The 'Model G' was 350 = 70 x 90 (346 actual) cc/cu, and there was no twin port model G version as far as I know.
Definatly a J2, the 350 model G had a different crank case (subtle) and the rear engine plate has an extra attacment point.
The frame is deeper than the G also, and it would have had tele forks as the head stock is slightly longer than the shorter pre war girder forked J2.
I would guess it would be 1949/50. If you want to buy parts, try
www.hitchcocksmotorcycles.com I've had a lot of success with them.