Correct. 61306 wasn't called "Mayflower" whilst in British Railways service. It was just 61306. But someone who bought it thought "Mayflower" was a nice name. Girlie.
It gets into Brighton at 7.40 and leaves about 10 I think. Earlier the same train is now en rooute around Surrey, it gets back to Victoria and then sets off for Brighton at 6 I think.
For the nerds, 61306 is a B1 class 4-6-0 designed by Edward Thompson of the LNER. Built by British Railways...