The Islamic State affiliate, made up of Sunni extremists, view Shiites as apostates.
The attack wounded more than 80 people, many of whom suffered severe burns.
More broadly, the Afghan capital has come under attack several times in 2017.
They took place during a ceremony to mark the 38th anniversary of the Soviet invasion in the country.
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Islamic State has expanded its presence in Afghanistan since it first appeared in the region in 2015.
According to them, there were at least two, maybe three suicide bombers who also used grenades in the attack.
The claim reflects witness reports that one bomber sneaked into the center and exploded his device. It also hit the Afghan News Agency, which has an office on the floor above, Afghan Interior Ministry spokesman Nasrat Rahimi said.
Aamaq in its statement also claimed the center was funded by Iran and propagating Shiite beliefs.
President Ashraf Ghani's spokesman issued a statement calling the attack an "unpardonable" crime against humanity and pledging to destroy terrorist groups, according to Reuters.
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Teigen, who called the ordeal "a flight to nowhere", tweeted that the unauthorised passenger "must be so mortified". The airline did not immediately say why the wrong boarding pass was accepted.
The interior ministry told the BBC the suicide blast was followed by another two explosions in the area.
August 25: 20 people killed in an attack on a Shiite mosque, for which ISIS claims responsibility.
"In one of the deadliest years on record, journalists and other civilians continue to be ruthlessly targeted by armed groups", the statement added.
Separately, Dawlat Abad District Gov. Mohammad Karim said a powerful mine killed six shepherd children ranging in age from 8 to 10 on Wednesday.
No one immediately took responsibility for the attack but Karim blames the Taliban, saying the insurgents planted the mine to target Afghan officials and security forces. Security in the city has been ramped up since May 31 when a massive truck bomb ripped through the diplomatic quarter, killing some 150 people and wounding around 400 others - mostly civilians.
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