The order base its beliefs on Guthix's principles, founded and made to fight against the gods. But after the god's death, the order lost its path and have since begun experimenting on local villagers in order to create a new god to worship.
The Order of Ascension most commonly refer to the six legiones who lead the Order. These six legiones are powerful mages using magic learned from Guthix. They used this magic to experiment on villagers kidnapped from local villages in order to try to "recreate" Guthix despite his wishes and the wishes of the Guardians of Guthix, specifically Ocellus Virius, the Guardian outside of the monastery.
According to the History of the Order, Gladii are the earliest members of the known Order, as they were first created in 1743 A.G.S (around Second or Third Age). The next member that was created was the Rorarii in 1895 A.G.S (5 years after the creation of the Legiones in the Third Age). Scutarii were created in 1935 A.G.S (around Third/Forth Age), making them the second most recent member. The "youngest" members of the Order are the Capsarii, made in 2017 A.G.S (around Forth/Fifth Age).
In the Sixth Age, the Order seems to be stagnated, as they have not made much progress in creating new "members" or finishing the ascended Guthix. Ocellus also mentioned that the Order had stolen Algarum thread and used it to create armor for themselves. Ocellus was aware of their progress and stopped it by stealing the thread from them.
During this time, a seventh legio, Legio Septimus, broke away from the Order with several Ascended, believing that his fellow members did not do anything but feed off the crystals littering the complex. He and his order later allied with Nomad to create an ascended Guthix of their own, which was defeated by the player and killed by Death/Icthlarin in Nomad's Elegy. While it is unknown what happened to Septimus (since the player had the choice of killing or sparing him), it is known that there are still several of his followers still lingering around in the Underworld in hiding.