To move a table from one filegroup to another while the system is "live" and "online", recreate its clustered index...  Be sure the target filegroup is already sized to handle the incoming data :)

ALTER DATABASE databaseName
MODIFY FILE
    (NAME = datafileLogicalName,
    SIZE = 2000MB);
GO

CREATE CLUSTERED INDEX [i_tableName_cu] ON [dbo].[tableName] 
(
[columName] ASC
)
WITH DROP_EXISTING
ON [filegroupName]
GO