Air carrier First Officer reported a communication error which resulted in a heading deviation after takeoff and an ATC low altitude alert. The flight was assigned a heading by ATC and continued the flight to the destination.

Narrative

While taxing out for departure from SAN; ground control advised us that we had an amendment to our clearance and we advised that we were ready to copy. the clearance that I was given was: At 400' fly heading 290 expect radar vectors to rejoin the ZZOOO4 departure; climb maintain FL190. But what I read back and had written down was on departure at 400' fly heading 190 expect radar vectors to resume the ZZOOO4 and to climb FL190. After I read back the incorrect clearance the ground controller responded with 'Aircraft X read back correct' We then continued our taxi to runway 27 and once we were airborne and had been passed to departure from the tower controller who had simply cleared us for takeoff without assigning any new headings or altitudes. We checked on with departure on the 190 heading and the controller responded with a low altitude alert and assigned us a 180 heading. Both the captain and I had 'terrain' selected on the ND prior to takeoff. We were later advised by the same departure controller that we had a possible pilot deviation and gave us a phone number to call. We asked what the deviation was for and he informed us that we were given a 290 heading to fly on departure not a 190 heading. we then copied the phone number and continued the flight without further incident.