Police had earlier put the toll at 58 presumed dead.
Victims of the fire have made clear their demands to Mrs May in the two-and-a-half hour meeting in Number 10 this afternoon.
Cundy gave Alhajali's name at the press briefing. Some said they had never seen a building fire advance so quickly.
"I spoke to the casualty helpline and they didn't have any information. they didn't offer any help".
Cundy warned that the toll of 79 people could change. "(But there was) nothing, they weren't collating these numbers". "It will also be a subject that the separate criminal investigation will be looking at", he said.
"It was a death trap, and they knew it", one person shouted as demonstrators surged inside the offices of the Kensington and Chelsea council, responsible for managing the social housing block in a working-class enclave of one of Britain's richest districts.
First Secretary of State Damian Green has defended the prime minister, saying she was as "distraught as we all are".
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"The response of the emergency services, NHS and the community has been heroic", she said in a statement.
Writing in the Observer newspaper, Khan says it may well be the "defining outcome of this tragedy that the worst mistakes" of that era become a thing of the past.
"At least she could've met the victims, [Labour leader Jeremy] Corbyn was a good man. he came and met the people", one victim told Sky News.
The government has promised a full public inquiry, but that has done little to a sense of frustration and anger among residents and neighbours who demand answers for how the blaze spread so quickly and trapped so numerous tower's roughly 600 residents.
May was criticised for avoiding locals when she visited the charred shell of the 24-storey tower on Thursday and faced cries of "Shame on you" and "coward" when she returned on Friday.
He said: "The conditions due to the fire damage verge on indescribable, which is why this will be such a lengthy operation taking weeks to complete".
There is simmering anger in the multi-ethnic north Kensington area hit by the blaze, and public fury has been directed at senior government figures, including May, who was jeered Friday after she visited.
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Residents say they were ignored and want the government to take responsibility.
"This is an absolutely terrible fire that took place". She added that "I have heard the concerns and I have ordered immediate action across the board to help victims' relatives and the survivors". "The country has witnessed a succession of awful tragedies". People have lost their lives and others have lost everything, all their possessions, their home and everything.
"But it is a bad tragedy".
"It's hard to describe my feelings, because I can not imagine, and I would not want to put myself in the position of those families who have lost their loved ones".
"What I'm now absolutely focused on is ensuring that we get that support on the ground".
It says that the Grenfell fire ranks above the fire at Bradford City Stadium in northern England on May 11, 1985, when flames swept through the wooden stands and killed 56 people. However, there may be other people who were in there on the night that others were not aware were there.
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Theresa May is chairing an emergency Cobra meeting this morning, and is expected to give a statement in Downing Street afterwards. An AFP reporter could see a helicopter and many emergency vehicles at the scene, which was closed off by a large police cordon.
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