The only authentic genealogy recounting the names is until the Prophet's ancestor Fihr. See here.
People have agreed that Fihr was descended from a person named Adnan, but an exact list of names is a matter of debate. Then, people have agreed that Adnan was desended from Ismail (AS), but the exact list of names is in even more debate. Since neither the Quran nor the Sunnah contains a recounting of the ancestry, it is hard to get something very authentic. Refer to the Sealed Nectar and its chapter on ancestry for the claimed versions.
We don't believe the Prophet (SAW) was a descendant of Ismail (AS) particularly due to some ancestry record. We believe it because he was a prophet from God and he told us so. And Allah told us so in the Quran:
And [mention] when Abraham was raising the foundations of the House and [with him] Ishmael, [saying], "Our Lord, accept [this] from us. Indeed You are the Hearing, the Knowing.
Our Lord, and make us Muslims [in submission] to You and from our descendants a Muslim nation [in submission] to You. And show us our rites and accept our repentance. Indeed, You are the Accepting of repentance, the Merciful.
Our Lord, and send among them a messenger from themselves who will recite to them Your verses and teach them the Book and wisdom and purify them. Indeed, You are the Exalted in Might, the Wise."