Subfactorial

Find a six-digit number represented as $\overline{abcedf}$ that has the following property:

$\overline{abcdef} = !a + !b + !c + !d + !e + !f$

The exclamation mark in front of the number indicates a sub-factorial which is defined as follows:

$!n = n! \left(1 - \frac 1{1!} + \frac 1{2!} - \frac 1{3!} + \cdots + (-1)^n \frac 1{n!} \right)$

For examples: $!2=1$ and $!6=265$

The exclamation mark after the number denotes the factorial function ; for example $8! = 1 \times 2 \times 3 \times \cdots \times 8$ .