Bob: I'm thinking of a whole number between 10 and 1300.
Alice: Is it less than 500?
Bob answers her, but he lies.
Alice: Is it a perfect square?
Bob answers her, but he lies.
Alice: Is it a perfect cube?
Bob answers truthfully.
Alice: If I knew whether the second digit from the left was a 1, I would know your number. Is it?
Bob answers her truthfully and Alice announces what she thinks is the number. But she is wrong.

What is Bob's number?