Corporate pilot reported receiving a low altitude alert from ATC during departure when the autopilot turned the aircraft to a fix that was incorrectly programmed. ATC vectored the aircraft back on course.

Narrative

Upon departing ZZZ; we were given takeoff clearance and departed. When we turned on the Auto Pilot; the airplane didn't do what was loaded in the box and I was late in recognizing it. We had briefed prior to departure what we were going to do; and I had verified the initial heading to the right in the GARMIN. However when the autopilot came on; we realized it wasn't responding correctly and were late to make the correction. At this time Tower asked to confirm we were on the appropriate departure; which was loaded as the ZZZZZ [departure]. We were issued a low altitude alert by the tower; but never received anything from the terrain warning. Then some vectors were given and we were handed over to Approach.Once we were enroute we noticed a fix that was above Runway XX which we believe was loaded accidentally; and I failed to recognize it while checking the flight plan page. I was able to read out all the proper fixes and missed the fix that was loaded above the departure runway. I believe this is what led to the improper actions by the autopilot.Additional distractions were we received a fuel imbalance prior to take off that we had rectified. I also double checked the heading and everything appeared normal on departure with proper headings verified. Moving forward we will be more vigilant in checking our GARMIN sequences on the flight plan page. We will also plan to hand fly this departure in the future to ensure it is flown properly with a call out at the altitude we make the turn to the left; so it does not happen again pre-maturely.