I’m not sure I’ve ever been to a musical that packs in as much story as “Dirty Business.” And, yes, I have seen “Les Miserables.”

The world-premiere musical at History Theatre is an ambitious affair that whips around between the mostly separate careers of four female spies during World War II. Minnesota native Betty Pack (Kendall Anne Thompson) is the central figure who should probably be more central. But that can’t happen because there’s also code breaker Elizebeth Friedman (Melanie Wehrmacher), spy master Vera Atkins (Jennifer Baldwin Peden) and entertainer/undercover agent Josephine Baker (Timotha Lanae).

Telling one of their stories would be a big job. But telling all of them, while skipping back and forth during the 1930-’42 time period, asks a lot of the 11-person cast, several of whom put on and take off as many accents as hats. It also asks a lot of the set, which projects images on oddly rumpled screens to establish the time period and country we’re in.

“Dirty Business” is best when it focuses on Pack. Thompson brings spunk and personality to the woman who announces early on, “I’m not a saint and I don’t care what you think of me.” Mostly, you’ll think that Pack might be a tough person to have as a friend (the show doesn’t have time to indicate whether she has any friends, in fact) but she’d be a lot of fun at a party, telling stories of the time she slept with a general to help her lover escape custody, or slept with a Nazi to distract him while someone cracked a safe, or slept with a diplomat to get secret codes.

The show has fun with Pack’s very particular set of skills in a song called “Honey in the Trap,” with the character breathlessly promising one dude, “You can trust me,” and then immediately handing his secret decoder machine to a confederate. (In her ruthlessness and drive, Pack may remind you of “Evita,” which “Dirty Business” occasionally resembles in scope and structure.)

With influences ranging from Gilbert and Sullivan to midcentury jazz, the songs by Robert Elhai and Laurie Flanigan Hegge (who also wrote the show’s book) are engaging pastiches, highlighted by Baker’s very Folies Bergère-esque “Ah Paris!” and the witty “The Roses of Warsaw,” a bit of patriotic schmaltz in the same vein as “The White Cliffs of Dover.”

What we don’t get from “Dirty Business” is a sense of the psychological cost of that dirty business. Did the burned bridges Pack left in her wake help her figure anything out? What made her so steely? What role did Baker’s blackness play in her capers? There’s something to be said for a show that leaves us wanting more. But especially in a musical that opens and ends with the exact same scene, we’d like a bit more insight into what makes the characters tick in between.