Galileo is a

Lingbleed, a true-breeding bay really means he can't get chestnuts. Grey is a sort of fading of pigmentation and is inherited by direct transfer from a grey parent. As it happens Galileo gets plenty of chestnuts, Australia being a recent example, and is not a true breeding bay.

Thanks formtwist

I think all Galileo's Epsom Derby winners have been chestnut.
Is there a link between the colour gene and others that make a horse by him excel?

A grey horse will have on or more grey parents.
Bhutan's dam Ecology is grey.

Look at Bhutan's pedigree here
http://www.pedigreequery.com/bhutan6

You can trace the grey colour from offspring to parent
Bhutan (grey), sire Unbridled's Song (grey), Trolley Song (grey), Caro (grey), Fortino (grey), Grey Sovereign (grey), Kong (grey), Baytown (grey), Princess Herodias (grey), Queen Herodias (grey), The Tetrarch (1911 (grey).

I think all grey colour horses trace to the Irish horse The Tetrarch.  I may be wrong.

Nijinsky got a few Derby winning chestnuts too - Shahrastani, Lammtarra, Ferdinand...and the very flashy chestnut Generous was a grandson.

Galileo's mother was chestnut, 3 of his brother/sisters were chestnut. As mentioned above about his Epsom derby winners, he also got these chestnuts

Alice Springs, Cape Blanco, Allegretto, The United States