A main conflict in the story is between the old, ordered, pre-World War II way of life and the literal leveling and destruction the Wormsley Common Gang see all around them in the aftermath of the war.
In the early 1950s, London had still hardly been rebuilt after the massive destruction of the repeated bombings it endured in World War II. One house, however, stands tall amid the bombed rumble: a carefully maintained architectural treasure designed and built by Christopher Wren, who also designed St. Paul's cathedral.
When T calls the house beautiful, which he has toured simply by asking the owner to see it, Blackie, the head of the gang, feels uneasy:
It was the word 'beautiful' that worried him—that belonged to a class world that you could still see parodied at the Wormsley Common Empire by a man wearing a top hat and a monocle, with a haw-haw accent.
The boys decide to vandalize and destroy this architectural treasure—and succeed in doing so.
This act represents multiple levels of conflict. By doing this, the boys put themselves in conflict with the law. They also put themselves in conflict with the ideals of the past. As Blackie's discomfort with the word "beautiful" indicates, they want to destroy that past. While not stated overtly, the implication is that the boys hold the values of the past responsible for the present devastation in which they live. The past has a deceptive and an ambivalent beauty that has to be destroyed. This destruction, however, puts the boys in conflict with themselves: they are destroying a house they also grudgingly admire.
As the above quote about the top hat suggests, the lower class boys additionally feel in conflict with the English class system that has oppressed them: in destroying the house, they symbolically attempt to destroy class distinctions. When the truck driver who unwittingly aids in the destruction laughs and thinks it is funny, this also represents a class conflict: the ascendant new working class doesn't understand the values of the old world.