An online cheat sheet to an NYPD promotion test gave applicants an unfair advantage when it came to passing the make-up test, nine sergeants allege in a new lawsuit.

The officers filed suit Thursday against the city in Manhattan federal court, a month after The Post exclusively reported that a cheating scandal was running rampant within the department – one where make-up test-takers passed at rates eight times higher than those who took the original exam.

In the lawsuit, the plaintiffs allege that sergeants who couldn’t attend the first go of the April 18, 2015 test – formally known as Lieutenant’s Promotional Exam No. 5535 — had two months to study questions and answers that were posted on cop blog “Rising Star” prior to the make-up date.

The blog contained a photo showing all of the “correct” answers – which were identical to an answer key eventually provided to cops during a “protest session” in which officers could complain about questions they think are unfair or erroneous, the suit says.

Of the 2,401 sergeants who took the initial April 18 test, only 164 – or 6.8 percent — passed, compared to 48 percent who passed of the 80 officers who took the makeup, court documents say.

The 80 sergeants who took the make-up test because they weren’t able to take it the first time included Jewish applicants who observe the Sabbath, others who were on military leave, and sergeants injured when the test was first offered. That group passed at a significantly higher rate, The Post previously reported.

The lawsuit also alleges that officers were free to cheat, if they wanted to. Exam proctors during the April test frequently exited the room, leaving it unsupervised and test-takers were permitted to go to the bathroom without having to sign out.

And officers were allowed to keep their cellphones and other belongings while taking the exam.

“We will review all the claims,” said a spokesman for the city Law Department.

An internal investigation into whether the testing process was corrupted is still ongoing, according to a police official.

“At the end of the day, you want to focus on the people who may have been involved in the wrongdoing and not penalize those who did nothing wrong,” said the official. “It’s a real dilemma here because you have to be fair to everybody.”

Randolph McLaughlin, who represents the lieutenants who are suing, said: “Cheaters shouldn’t be promoted. There are lots of possibilities for cheaters here. Given that, we shouldn’t have anyone promoted to lieutenant until the NYPD figures out who the cheaters were – if they can.”

On Monday, McLaughlin will be in court to ask a federal judge to bar the NYPD from making promotions pending the department’s investigation.