The group says it has 1.5 terabytes of the company's data and has posted episodes of Ballers and Room 104 online.
It added that more material would be released "soon".
HBO confirmed it had experienced a "cyber incident" in a statement.
In an email published by Entertainment Weekly, the hackers appeared to offer more details in exchange for favourable coverage.
"Hi to all mankind," they wrote. "The greatest leak of cyber space era is happening."
They encouraged recipients to download the material and added: "Whoever spreads well, we will have an interview with him."
Reports have said the allegedly stolen Game of Thrones script appears to be from the fourth episode of season seven, which is currently being broadcast.
The BBC has not been able to independently verify that the hackers possess the material they claim to have stolen.
HBO confirmed that a "cyber incident" had resulted in the compromise of information.
"We immediately began investigating the incident and are working with law enforcement and outside cybersecurity firms," the firm added.
"Data protection is a top priority at HBO, and we take seriously our responsibility to protect the data we hold."
The intrusion was "obviously disruptive, unsettling, and disturbing for all of us," said chairman and chief executive Richard Plepler in an email to HBO employees.

The Friends of Sherwood Forest say the Â£5.3m centre, located on the edge of a country park, will devastate the area.
"It has all sorts of wildlife - insect life and newts," the charity's group spokesman Adrian Wilson said.
Nottinghamshire County Council said the new site would "balance the interests" of the community and visitors.
The current centre is in Sherwood Forest Country Park, on land designated as a site of special scientific interest but plans are to move it to Forest Corner, on grassland on the edge of the park.
The council has signed a contract with the Royal Society for the Protection of Birds (RSPB) to build and run a new one.
County council spokesman Derek Higton said: "The centre will be run by one of the world's leading conservation organisations - and that's why we partnered with them."
Updates on this story and more from Nottinghamshire
"We are confident we can balance the interests of the community and the interests of the 350,000 people who visit Sherwood Forest and Major Oak every year."
Detailed plans have not yet been submitted.
Mr Wilson said: "It is priceless - if you were to drive up the road, on a Sunday lunchtime, you would see the cricketers out here all in their whites, people sat round watching the cricket match.
"You get that feeling straight way - this is England."
Shawn and Linda Geery, who walk their dog in the country park, said: "It is too near the edge of the forest, the parking problems will be horrendous."
Construction is scheduled to start in December 2016 with the opening expected by winter 2017.

Sir Gareth attended a ceremony at Windsor Castle on Thursday, after being named in the Queen's Birthday Honours list in June.
The 68-year-old former scrum half won 53 caps for Wales from 1967 to 1978.
He also won 10 caps for the British Lions' winning series in New Zealand and South Africa.
At 20 he became Wales's youngest captain, and during his era the Welsh side dominated the Five Nations Championship
Originally from Gwaun-Cae-Gurwen in the Swansea Valley, Edwards spent his playing career with Cardiff RFC.
In 1974 Edwards was named BBC Wales Sports Personality of the year.
After his retirement in 1978, he became Captain on the popular sports quiz, Question of Sport.
He now works as a pundit for both the BBC and S4C.
He is married to his childhood sweetheart Maureen and they have two sons, Owen and Rhys.

Officers believe the local man, who was working at Fruehauf on Houghton Road, fell from a cherry picker at about 13:45 GMT on Thursday.
Lincolnshire Police said it was carrying out a joint inquiry with the Health and Safety Executive.
The company was cooperating fully with the investigation, the force added.
A group of hackers claims to have stolen the script for a forthcoming Game of Thrones episode and other data in a breach at entertainment firm HBO.
[NEXT_CONCEPT]
A new visitors' centre proposed for Sherwood Forest will ruin "a priceless" tract of land and threaten wildlife, opponents say.
[NEXT_CONCEPT]
Wales rugby great Gareth Edwards has been knighted by the Duke of Cambridge in recognition of a glittering sporting career and services to charity.
[NEXT_CONCEPT]
Police are investigating the death of a 64-year-old man who died after an industrial incident in Grantham.