It appears that there are several more units in Gaza than I thought when writing my original post.
Apparently the Givati Brigade is operating inside the Gaza Strip and not simply securing the Israeli side of the border. This became evident when the IDF spokesperson published a short video clip of a booby trapped housed the brigade found near a UN school in Gaza. The Givati, nicknamed ‘Samson’s Foxes’, have a solid history of operating in Gaza, and I was somewhat surprised when it first seemed that they would not take part in this operation. Exact place of operation remains unknown.
The Sayeret Maglan special forces unit also appeared inside the Strip in a tweet by IDF spokesperson. Their presence was perhaps less of a surprise than the fact that their presence was confirmed openly by the IDF.
The 7th Armored Brigade is also apparently in full use in the central regions of the Gaza Strip. The brigade is best known for its stubborn defence in the opening hours of the Yom Kippur War, when in spite of large losses it managed to stop the vastly larger Syrian force trying to force its way through the northern parts of the Golan Heights. Then-battalion commander Avigdor Kahalanis account of the battle, “The Heights of Courage”, is widely seen as a modern classic of tank warfare.
All in all, there now seems to be at least three infantry brigades (Golani, Nahal, Givati), two armored brigades (7th and 401st), as well as a single paratrooper brigade operating inside the Gaza Strip. It seems that both armored brigades are operating in the central area of the strip, where the terrain is more open, with the 401st being the northernmost of the two.
Update 30th of July:
The location of Nahal and the paratrooper brigade has become somewhat clearer. Nahal is operating in the northern parts of the Gaza Strip, according to this video released by the IDF.
The paratrooper brigade, on the other hand, is operating in Khan Yunis, according to a Haaretz interview with the brigade commander.
…And perhaps the last of IDF’s more well-known brigades, the 188th ‘Barak’ armored brigade, is also in the strip (or at least their infantry), bringing the toal up to three armored brigades.
Update 1st of August:
The 188th seems to be the southernmost of the armored brigades, operating around the northern outskirts of Khan Yunis. This is based on the fact that their presence in Eshkol has been confirmed, as it was there five of their soldiers who were killed by a mortar strike earlier this week.
Note that with the figure for IDF personnel killed in action already over 60 and climbing, it is over halfway to the corresponding number of losses IDF suffered during the Second Lebanon War.