Hundreds of names, addresses, birthdates and Medicare details were being posted under "good-list" and "naughty-list" on a blog belonging to the group.
The hackers had demanded a ransom to stop them from releasing the data but Medibank earlier this week said it would not pay it because it would encourage further crime.
Shortly after midnight, the group posted the first lists.
"Looking back that data is stored not very understandable format (table dumps) we'll take some time to sort it out," they said in the early hours of Wednesday.
"We'll continue posting data partially, need some time to do it pretty."
The hackers also appeared to have revealed screenshots of private messages recently exchanged between themselves and Medibank representatives.
Medibank said in a statement on Wednesday the files appeared to be "a sample of the data that we earlier determined was accessed by the criminal".
"This data includes personal data such as names, addresses, dates of birth, phone numbers, email addresses, Medicare numbers for ahm customers (not expiry dates), in some cases passport numbers for our international students (not expiry dates) and some health claims data," the insurer said.
"We will continue to work around the clock to inform customers of what data we believe has been stolen and any of their data included in the files on the dark web and provide advice on what customers should do."
Medibank said it expected the criminals would continue to release files on the dark web.
"We unreservedly apologise to our customers," Medibank CEO David Koczkar said.
"This is a criminal act designed to harm our customers and cause distress.
"We take seriously our responsibility to safeguard our customers and we stand ready to support them."
Medibank has previously confirmed almost 500,000 health claims were stolen, along with personal information, when the unnamed group hacked into its system weeks ago.
Some 9.7 million current and former customers have been affected.
No credit card or banking details were accessed.
Assistant treasurer Stephen Jones blasted the hackers on Wednesday.
"They're scumbags, they're crooks, they're criminals and we shouldn't be paying ransom," he told Sky News.
"We shouldn't be giving in to these fraudsters. The moment we fold it sends a green light to scumbags like them throughout the world that Australia is a soft target. We cannot give in and we won't give in."
Mr Jones said Australia needed to quickly lift protection against cyber threats.
Medibank is certainly not alone in refusing to pay a ransom demand, with a recent report finding 19 per cent of Australian companies responded to ransomware attacks by paying the fee.
Mimecast's 2022 State of Ransomware Readiness report found 20 per cent of companies were asked to pay between $500,000 and $999,999 for their information
Some 13 per cent of the businesses surveyed said the total cost of the ransomware attacks they'd experienced was between $1 million and $2 million.
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