Hey Sebastian, I'm not 100% sure myself, but my experience has been that it wont power up unless you feed it 2A. I use mine for portable projects, and have found that unless my battery supply can feed 2aH it wont boot.
The Galileo Gen 2 works with a power supply inside the range of 7V to 14V, about current with 1.5A you should be able to power the board. Without a WiFi card connected the power consumption is approximately 2.67W . We have also measured the power consumption with a Centrino Wireless-N 135 connected and it was around 3W. During those measurements we did not have any shield connected but if you add a shield the power consumption will increase by the amount of power that the shield would withdraw.
Thanks for the replies.
We have been using the kit adapter now (12VDC 1,25A) without any issues, the only extra "thing" connected to the Galileo board is an Telit HE910 mPCIe, and I have not seen any power related issues.