#250002

order	512
nilpotency class	3
isoclinism	isoclinic class of #249943
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#525	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,12)(4,18)(5,21)(6,10)(7,11)(8,15)(9,19)(13,17)(14,23)(16,24)(20,25)(22,26)(27,29)(28,31)(30,32)(33,34,38,42)(35,39,45,52)(36,40,47,54)(37,48,49,41)(43,55,57,50)(44,58,59,51)(46,53,60,61)(56,63,64,62);(1,3,7,17)(2,8,11,24)(4,14,20,30)(5,16,6,15)(9,23,26,32)(10,13,21,12)(18,27,25,28)(19,31,22,29)(33,35)(34,39)(36,43)(37,44)(38,45)(40,50)(41,51)(42,52)(46,56)(47,57)(48,58)(49,59)(53,62)(54,55)(60,64)(61,63);(1,2)(3,13)(4,9)(5,10)(6,21)(7,11)(8,15)(12,17)(14,28)(16,24)(18,19)(20,26)(22,25)(23,31)(27,30)(29,32)(33,36,38,47)(34,40,42,54)(35,43,45,57)(37,46,49,60)(39,50,52,55)(41,53,48,61)(44,56,59,64)(51,62,58,63);(1,4,7,20)(2,9,11,26)(3,14,17,30)(5,22,6,19)(8,23,24,32)(10,25,21,18)(12,27,13,28)(15,31,16,29)(33,37,38,49)(34,41,42,48)(35,44,45,59)(36,46,47,60)(39,51,52,58)(40,53,54,61)(43,56,57,64)(50,62,55,63)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157198);