Is the z-pack related to or in the same class as erythromycin?

YES: They are in the class of macrolide antibiotics. Z-Pak is "Azithromycin."

Yes: A Z pack is five days of azithromycin which is in the macrolide category of antibiotics, like erythromycin. Interestingly, in my experience most people who claim an allergy to erythromycin describe stomach upset as their allergy symptom. This is not an allergy, this is a side effect to erythromycin, perhaps its most common one. Azithromycin causes some stomach upset but less that erythromycin.