Building officials have not commented since the fire.
May was accosted by furious protesters earlier today as she attended the nearby Kensington Town Hall.
Many are demanding answers for how the blaze spread so quickly.
"I really hope it won't", but Commander Stuart Cundy admitted the number of casualties may increase. This 58 include the 30 already confirmed dead, according to Cundy. "It may well be the defining outcome of this tragedy that the worst mistakes of the 1960s and 1970s are systematically torn down", he said. British health authorities say that 19 fire survivors are still being treated at London hospitals, and 10 of them remain in critical condition.
Cundy said the police investigation would look into the building's 2016 refurbishment and promised to prosecute "if there is evidence".
Cundy said police had now managed to get to the top of the tower and had undertaken a first visual search for victims, ahead of later painstaking searches.More news: 2nd LD: Protesters storm local town hall over London fire
"The work to search the building is challenging, but naturally could never be done quickly enough for those now having to live with the uncertainty of knowing where their loved ones are". "Sadly, we do not expect any survivors". They say their complaints were ignored - and fear it was because the tower housed mainly poor people in a hugely wealthy neighbourhood.
She said support on the ground for families immediately after the blaze was "not good enough" and ordered that more staff be deployed at the scene in high-visibility clothing.
Officials said due to the nature of their deaths, it's possible all of the victims of the fire might not be conclusively identified. They say they will also use visual elements like tattoos and scars, in the painstaking process.
In a message released on her official birthday, she said people have been "resolute in the face of adversity" following recent tragedies in London and Manchester. Earlier in the day, the 91-year-old monarch described the country's mood as "somber" but insisted that Britain remained resolute during a hard time.
British Prime Minister Theresa May will meet with survivors of the London high-rise fire at her Downing Street office. The building was gutted in a blaze early Wednesday morning that has also left dozens missing and hundreds of others homeless.
Some of the anger stems from reports that Grenfell Tower had no central sprinkler system and was built with flammable materials that residents told the government needed to be fixed.