#502021

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33611	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,24)(12,32)(13,38)(14,41)(15,21)(16,22)(17,28)(18,27)(19,26)(20,44)(23,42)(25,56)(29,51)(30,60)(31,68)(33,54)(34,57)(35,70)(36,58)(37,47)(39,50)(40,46)(43,45)(48,67)(49,61)(52,62)(53,66)(55,63)(59,64)(65,72)(69,71);(1,3,4,7)(2,5,6,8)(9,11,25,61,16,31,14,28,15,29,44,34,42,30,43,67)(10,17,45,68,23,49,20,24,21,48,41,60,22,57,56,51)(12,33,62,65,37,66,35,13,27,63,71,40,36,26,64,39)(18,50,70,19,47,46,52,55,32,38,59,53,58,72,69,54);(2,6)(5,8)(9,12)(10,18)(11,26)(13,34)(14,35)(15,27)(16,37)(17,46)(19,51)(20,52)(21,32)(22,58)(23,47)(24,53)(25,62)(28,65)(29,66)(30,63)(31,33)(36,42)(38,49)(39,61)(40,67)(41,59)(43,64)(44,71)(45,70)(48,72)(50,57)(54,60)(55,68)(56,69);(9,13,25,63,16,40,14,26,15,39,44,33,42,65,43,66)(10,19,45,46,23,55,20,38,21,53,41,72,22,54,56,50)(11,27,61,71,31,36,28,64,29,12,34,62,30,37,67,35)(17,47,68,52,49,32,24,59,48,58,60,69,57,18,51,70)

Polycyclic presentation:

2^9|b^2=efh,d^2=ehi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409217);