I have seen the number quoted as > 650 billion. The quote is from a Danish mathematician, Soren Ehlers. Intuitively, this does not seem correct to me. Any mathematicians/statisticians out there who can help either verify the huge number, or give me a correct number?
To expand on Zhaph's answer, I'm assuming that you have a fixed set of six colors, and each block is one color.
If that's the case, this is a simple combination problem on top of the orientation problem. Since there are six colors and six bricks, there are 6! ways to assign those six colors to the six bricks, or 720 ways.
Now, we apply this combination to our orientation, and get:
915,103,765 orientations x 720 color combination/orientation = 658,874,710,800 combinations
I'm not sure what the rules are around the colours, however from the LEGO Facts page:
Take six eight-stud LEGO bricks (2x4) – how many ways can they be combined?
With the aid of computers, the exact number of combinations has been calculated as 915,103,765!
Just so you know, two eight-stud LEGO bricks can be combined in 24 different ways and three eight-stud LEGO bricks in 1,060 ways.
This is probably based on the work done here.
The colour combinations would therefore be higher than that - but you'd need to be avoiding duplicate combinations.