Book about boy who suddenly attains powers, goes to school for kids with powers

It's actually a fairly well known book (for book readers anyway), because there's a lot of similarities between it and Harry Potter. The book's name is Wizard's Hall.

It's possible you're thinking of Jenny Nimmo's Children of the Red King series. With enough lateral thinking, it matches all of what you mentioned in the question:

a boy who gains powers all of a sudden

In the first book, Charlie Bone (the protagonist) learns unexpectedly that he has a magical ability ...

He goes to a special school,

... which means he is sent to Bloor's Academy, a unique school dedicated to ...

and he meets people who have other types of powers;

... children with endowments: the word used in the series for the magical powers inherited by some descendants of the semi-legendary Red King.

for instance, one of his friend had perfect photographic memory,

By simply looking at a photograph, Charlie can hear the voices of the people in it at the time the photo was taken, as though he was actually there. This is his endowment. Sometimes he even manages to enter into the photo and interact with the people in it.

and there is a girl.

There's more than one girl in the series! You may be thinking of Olivia, a good friend of Charlie's who at first doesn't seem to be endowed but later discovers her latent powers, or the mysterious Emila Moon (Emma Tolly).

I think it talks a little about Charlemagne,

Here you may be mixing up the character of the Red King with the name of Charlie Bone.

and he tries to learn about his past.

As in many fantasy novels, the protagonist learning about his own past is a key plot point. In particular, Charlie's father went missing years before the series started, and the quest to find out what really happened to him is an important part of the story.

I think The Paladin Prophecy is the book you are referring to.

Will West is careful to live life under the radar. At his parents' insistence, he's made sure to get mediocre grades and to stay in the middle of the pack on his cross-country team. Then Will slips up, accidentally scoring off the charts on a nationwide exam.

Now Will is being courted by an exclusive prep school . . . and is being followed by men driving black sedans. When Will suddenly loses his parents, he must flee to the school. There he begins to explore all that he's capable of--physical and mental feats that should be impossible--and learns that his abilities are connected to a struggle between titanic forces that has lasted for millennia.

Co-creator of the groundbreaking television series Twin Peaks, Mark Frost brings his unique vision to this sophisticated adventure, which combines mystery, heart-pounding action, and the supernatural.

The sinister organization at the school are the "Knights of Charlemagne.".