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The city is moving to shut down a Staten Island strip club where it says customers could get more than just lap dances, according to legal papers.

Touch Gentlemen’s Club on Victory Blvd. near Travis Ave. in Travis proved a bit too true to its name, according to a nuisance-abatement lawsuit filed Monday.

Twice in the past three months, an undercover cop busted prostitutes inside the club — one of whom offered sex for $1,500, the suit alleges.

The club’s liquor license is held by Anthony Perosi III, who in 2015 split a $136 million Powerball jackpot with his dad.

Perosi called the allegations in the lawsuit “nonsense” and referred all questions to his lawyer.

Touch has been the scene of violence — including shootings, both nonfatal, on Feb. 27 and April 28.

In addition, a police officer was dragged by a vehicle outside the place on June 3, and a patron was beaten so badly on June 26 that his jaw and nose were broken, the suit alleges.

The club has changed hands over the years, operating under the names Pulse, Lipsticks and, until last year, Hush.

The city suit refers to the owner of the property as John Doe and Jane Doe.

On May 6, court papers say, a 37-year-old prostitute inside the club agreed to perform oral sex on an undercover cop for $200. On June 17, a 29-year-old Bronx woman offered an undercover sex for $1,500, the suit alleges. Both were arrested.

Police also sent in an undercover cop on June 1 and arrested a john who tried to pay her $20 for oral sex, according to the suit.

The city’s lawsuit is looking for an order to force the club to shut down for a year and pay $1,000 for each day it “intentionally conducted, maintained or permitted the public nuisance.”

Club manager Thomas Wolf said Touch operates above-board.

“A lot of the things they’re coming up with really had nothing to do with us,” he said.

Touch helped police investigate the June 3 dragging incident, he said, and the dancers are informed prostitution won’t be tolerated.

“They’re told when they’re hired that it’s a strip club, not a brothel,” Wolf said.

The city has also filed suit against Anthony Sciandra, who owns the property where the strip club sits.

He declined comment Monday, stating he hadn’t heard about the lawsuit.

“I’ll have to contact my lawyer,” he said. “I’m just learning all this on the phone now.”