The 405th is based out of Diego Garcia. They arrived on scene during the Covenant attack of Mombasa, Kenya on Earth. They went on to fight on the city streets of New Mombasa valiantly.
The 405th assisted the Master Chief in the destruction of a Scarab that was moving through the city. Members of this unit then traveled to Installation 05 along with the Chief aboard the UNSC In Amber Clad.
So....they exist, but aren't a Spartan group.
*insert 20 minute gap*
Actually, I couldn't find anything that described the unit they belonged to. Considering that there were only 33 of them in combat duty, and were generally kept together and moved as a unit (starting off), I bet they were their own unit.