You need a permit to purchase a handgun or an assault weapon...
https://www.revisor.mn.gov/statutes/?id=624.7131
Here is the definition of an assault weapon...
Bureau of Criminal Apprehension List of Firearms
August 1, 2016
As required by Minnesota Statutes, section 624.712, subdivision 8, the following is the current authoritative list of firearms included within the definition of “semiautomatic military-style assault weapon.”
A firearm is considered to be a semiautomatic military-style assault weapon, if it is:
(1) Any of the following firearms, or copies or duplicates of the firearms, in any caliber, known as:
(a) Avtomat Kalashnikov (AK-47) semiautomatic rifle type;
(b) Beretta AR-70 and BM-59 semiautomatic rifle types;
(c) Colt AR-15 semiautomatic rifle type;
(d) Daewoo Max-1 and Max-2 semiautomatic rifle types;
(e) Famas MAS semiautomatic rifle type;
(f) Fabrique Nationale FN-LAR and FN-FNC semiautomatic rifle types;
(g) Galil semiautomatic rifle type;
(h) Heckler & Koch HK-91, HK-93, and HK-94 semiautomatic rifle types;
(i) Ingram MAC-10 and MAC-11 semiautomatic pistol and carbine types;
(j) Intratec TEC-9 semiautomatic pistol type;
(k) Sigarms SIG 550SP and SIG 551SP semiautomatic rifle types;
(l) SKS with detachable magazine semiautomatic rifle type;
(m) Steyr AUG semiautomatic rifle type;
Street Sweeper and Striker-12 revolving-cylinder shotgun types;
(o) USAS-12 semiautomatic shotgun type;
(p) Uzi semiautomatic pistol and carbine types; or
(q) Valmet M76 and M78 semiautomatic rifle types.
(2) A semiautomatic rifle that has an ability to accept a detachable magazine and has at least two (2) of the following:
(a) Folding or telescoping stock;
(b) Pistol grip that protrudes conspicuously beneath the action of the weapon;
(c) Bayonet mount;
(d) Flash suppressor or threaded barrel designed to accommodate a flash suppressor;
(e) Grenade launcher.
(3) A semiautomatic pistol that has an ability to accept a detachable magazine and has at least two (2) of the following:
(a) Ammunition magazine that attaches to the semiautomatic pistol outside of the pistol grip;
(b) Threaded barrel capable of accepting a barrel extension, flash suppressor, forward handgrip or silencer;
(c) Shroud that is attached to or partially or completely encircles the barrel and permit the shooter to hold the semiautomatic pistol with the non-trigger hand without being burned;
(d) Manufactured weight of 50 ounces or more when the semiautomatic pistol is unloaded;
(e) Semiautomatic version of an automatic firearm.
(4) A semiautomatic shotgun that has at least two (2) of the following:
(a) Folding or telescoping stock;
(b) Pistol grip that protrudes conspicuously beneath the action of the weapon;
(c) Fixed magazine capacity in excess of five (5) rounds;
(d) Ability to accept a detachable magazine.
