Nicholson, who joined Enterprise straight out of the University of Missouri 32 years ago, becomes the first non-member of the Taylor family to head Enterprise, a private company located in St. Louis. Enterprise was founded by Jack Taylor, Andy Taylor’s father, in 1957 as a leasing operation. The two Taylors, Jack and Andy, went on to make Enterprise the runaway leader in the off-airport rental business and then a powerhouse at the airport as well.

Last year, Enterprise had $15.4 billion in revenues. Were it a public company, Enterprise would have ranked 181 on the latest Fortune 500 list.

The number of companies in that assembly that are headed by women has been growing, but is still only 20. In the world of large private companies the prevalence of women CEOs is also small. So Nicholson, in becoming a big-company CEO, is joining a select group.

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Andy Taylor has always made it clear that he wishes Enterprise to remain private and be a multigenerational family company. The generation of Taylors just below him include only women, one of whom, his daughter Christine Taylor-Broughton, 37 — who is called “Chrissy” — has just assumed the key Enterprise job of heading North American operations.

Both Nicholson and Taylor-Broughton began as management trainees at Enterprise. But the term, while absolutely accurate, somewhat glorifies the job, which sets new college graduates to running Enterprise branches and, among other duties, washing its cars.

Enterprise, which has 74,000 employees, believes itself to be one of the largest hirers of newly-minted college graduates in the U.S. Its current hire rate is more than 8,000 graduates a year. The two figures imply a large turnover, composed of degree-holders who don’t make it in Enterprises’s hard-working, incentive-based culture or don’t care for washing cars or both.

In 2007, when Nicholson was serving as chief operating officer of Enterprise, the company took the very major step of expanding from its off-airport empire into the highly competitive airport business, where its market share was only about 8%. Enterprise made its move — in the face of the looming financial crisis — by buying two airport brands, National and Alamo. The acquisitions, plus Enterprise’s own also-ran business, instantly jumped the company to an airport market share of about 27%.

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Enterprise has since heavily promoted National as a business brand (with ads that are directed to the “business pro”) and marketed Alamo as a leisure brand.

Official figures for airport market share do not really exist, and the subject has become more impenetrable as the rental-car industry has been overtaken by a merger boom. But Brian Johnson, a Barclays analyst, published market-share figures a few months ago that showed Hertz (HTZ) (after its acquisition of the Dollar and Thrifty brands) and Enterprise neck and neck in the airport race. Johnson gave Hertz (a stock pick of his) a 36% market share and Enterprise 34%. Avis (CAR), including its Budget brand, was third with 27%.

Nicholson told Fortune recently that she expects to focus hard on building Enterprise’s international business. In the midst of Europe’s economic malaise, Enterprise has negotiated a presence in two new countries, France and Spain, and worked at cementing its position elsewhere in the continent.