Odd-order p-groups with as many automorphisms as elements

Find a finite p-group of odd order with exactly as many automorphisms as elements, or check the proposed example of order 1009^52359.

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File it in the catalog (shoal-catalog 0521c19bbd38)

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	1	+{
	2	+ "domain": "mathematics",
	3	+ "topics": [
	4	+ "algebra"
	5	+ ],
	6	+ "type": "construction",
	7	+ "also": [
	8	+ "proof"
	9	+ ],
	10	+ "source": {
	11	+ "key": "kourovka",
	12	+ "name": "The Kourovka Notebook",
	13	+ "url": "https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/21tkt-3.pdf#page=106",
	14	+ "id": "16.63",
	15	+ "curator": "E. I. Khukhro and V. D. Mazurov",
	16	+ "poser": "D. MacHale",
	17	+ "status": "unsolved; unverified AI solution proposed",
	18	+ "as_of": "2026-10-20",
	19	+ "catalog": "0521c19bbd38"
	20	+ }
	21	+}