Twenty-five anti-bailout rebels from Syriza broke away to form the party on Friday.
Under Greece's constitution, the three biggest parties in parliament are given a three-day mandate to form a government following the resignation of Alexis Tsipras as prime minister.
The conservative New Democracy party currently has the mandate, which would pass to the next group if it was unsuccessful in its effort.
"We will take the mandate and we will utilize it," said Dimitris Stratoulis, a lawmaker with the Popular Unity party.
Greek opposition leader says to do all to avoid elections
Greece's conservative opposition leader Vangelis Meimarakis said on Friday he would try to cobble together a government with parties from the current parliament to avoid a negative impact from early elections.
Meimarakis, who heads the New Democracy party, was formally handed a three-day mandate by the country's president to try to form a new government earlier on Friday after Alexis Tsipras stepped down as prime minister on Thursday.
He told reporters snap elections expected on Sept. 20 "had no use" and he would try to form a government "in order to avoid, at this moment, all the negative effects that this election could create for a very long time."
Greek equities moved lower in early Friday deals after the leftist-led government's resignation, adding to the previous day's 3.5 percent drop with blue chip shares underperforming the broader market.
Blue chip stock index futures reversed opening gains of 5.9 percent with the September contract losing 0.76 percent and the underlying index of 25 stocks shedding 1.9 percent.
Prime Minister Alexis Tsipras resigned on Thursday, hoping to strengthen his hold on power in snap elections after seven months in office during which he fought Greece's creditors for a better bailout deal.
"The market started discounting the snap elections scenario on Thursday, we are seeing relatively lower pressure today on thin volume," said analyst Vasilis Andreou at Athens-based Berta Securities.
"The lack of clarity on the date of the vote is adding to the uncertainty," he said.
The new election, seven months after Tsipras swept to power on now-broken promises of ending lender-imposed austerity, injects renewed uncertainty into Greek policies.
Tsipras had long been expected to seek early elections in the autumn. But he was forced to move quickly after nearly a third of Syriza lawmakers refused to back Greece's new bailout programme in parliament last week, robbing him of his majority.
The Athens bourse's broader general share index was down 1.3 percent with banks losing 4.4 percent. The general index is down 21 percent year-to-date.
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