Brainteaser 1757: Enigma variations

From The Sunday Times, 19th May 1996 [link]

I recently joined Enigma, the international club for puzzle-solvers. When I received my membership card I noticed that my membership number was a six-figure number and that its digits could be rearranged to make three consecutive two-figure numbers.

I also noticed that its six digits could be rearranged to form two consecutive three-figure numbers.

Furthermore, no one with a lower six-figure membership number could make the same claims.

What is my membership number?

This puzzle was included in the book Brainteasers (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

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