The official HP Spare# is either going to be:
* 593553-001(47Wh 2.20Ah)
* 593554-001 (55Wh 2.55Ah)
These generally came with the 47Wh from the factory and they always seem to crap out the soonest. You’re better off with the 55Wh battery if at all possible. It is known as the HP MU06. It looks like HP no longer makes an OEM replacement, so you will need to get a good aftermarket one. Get one from a reputable company like [https://www.batteriesplus.com/battery/laptop/hewlett-packard/pavilion-g6-series/pavilion-g6=1b59wm-special-edition|Batteries Plus] to avoid future TSA headaches.