Kevin Andrews was repeatedly asked during a television interview with the Australian Broadcasting Corporation late yesterday to identify the IS chief -- widely seen as Abu Bakr al-Baghdadi.
But he constantly avoided answering directly.
I'm not going to go into operational matters obviously," he insisted, to which the interviewer replied: "I don't think it's operational, I think it's a matter of public record."
She added: "Minister, you're responsible for putting Australian men and women in harm's way in the cause of this mission, I'm surprised that you can't tell me the name of Islamic State's leader.
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Andrews insisted that IS was "a combination of groups".
"It's not just one person involved, there's a series of people involved and we must ultimately destroy all of them if we're going to degrade their operations in that area," he said.
Andrews later took to Twitter and said: "Focusing on individuals ignores the threat that extremist organisations present."
His refusal to name the IS chief was taken by Australian media to mean Andrews did not know who Baghdadi is.
It came just hours after he jointly announced with Prime Minister Tony Abbott that 330 non-combat troops were heading to Iraq for two years.
Some 170 Australian special forces are already in Iraq helping train government troops. Eight F/A18s based in the United Arab Emirates are also taking part in air strikes against IS militants.
The announcement came as the US, which is leading an air campaign against Islamic State, said the jihadists had lost control of "25 to 30" per cent of the territory it holds in Iraq after coalition air strikes and an Iraqi offensive.