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Mr Minnikhanov confirmed the deaths of seven children - four boys and three girls - as well as a female teacher and a female school worker at School No 175. It is believed the children were eighth graders (aged around 15).
Eighteen children and three adults were being treated in hospitals for gunshot wounds, fractures and bruising, Mr Minnikhanov added.
Footage shared on social media shows some children jumping from windows to escape as well as injured people being evacuated. Russian TV reported that two of the children had died after jumping from a second-floor window.
Mr Minnikhanov said the suspect was a registered firearm owner. According to Russian MP Alexander Khinshteyn, the suspect received a firearms licence only recently. The MP said it was for a semi-automatic shotgun.
Officials did not comment on the possible motive for the attack although Mr Minnikhanov called the attacker a "terrorist".
The suspect went on social media before the attack, posting that he planned to kill a "huge number" of people before shooting himself. Screenshots of his account, now blocked, show him wearing a black mask with the word "god" written across it in red.
A video on social media captured a teenager lying on the ground apparently being detained outside the building.
Kremlin spokesman Dmitry Peskov said President Putin had ordered the chief of Russia's National Guard to "work out as a matter of urgency new regulations on the types of weapon that can be in civilian circulation and that can be owned by the public".
The instruction was issued "given the type of firearm used by the shooter", Mr Peskov said. "The fact is that sometimes types of small arms are registered as hunting weapons, which in some countries are used as assault rifles," he explained
It is worth mentioning that the Kazan gunman allegedly used a gun very similar to one the attacker in Crimea used in 2018. It is a relatively cheap but powerful rifle which can be bought for the equivalent of just $280 (£200). There are a lot of discussions now in Russia about whether the regulations for civilian gun ownership should be changed.