#501850

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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	#33527	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

647 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)(11,18)(12,27)(13,17)(14,21)(15,36)(16,22)(19,28)(20,31)(23,43)(24,52)(25,38)(26,45)(29,58)(30,59)(32,46)(33,39)(34,40)(35,44)(37,49)(41,51)(42,53)(47,61)(48,60)(50,65)(54,72)(55,70)(56,64)(57,68)(62,63)(66,69)(67,71);(1,3,4,7)(2,5,6,8)(10,17)(11,23)(13,31)(14,15)(18,41)(19,27)(20,21)(22,40)(24,50)(25,56)(26,35)(28,32)(29,37)(33,34)(36,39)(38,43)(42,65)(44,64)(45,63)(46,49)(47,58)(48,59)(52,66)(54,55)(57,69)(60,61)(67,72)(68,70);(2,6)(5,8)(9,11,16,26)(10,18,22,45)(12,24,30,55)(13,25,34,56)(14,35,15,23)(17,38,40,64)(19,42,48,67)(20,41,39,63)(21,44,36,43)(27,52,59,70)(28,53,60,71)(29,54,37,50)(31,51,33,62)(32,57,61,69)(46,68,47,66)(49,65,58,72);(9,12)(10,19)(11,24)(13,32)(14,29)(15,37)(16,30)(17,27)(18,42)(20,49)(21,46)(22,48)(23,50)(25,57)(26,55)(28,31)(33,60)(34,61)(35,54)(36,47)(38,52)(39,58)(40,59)(41,65)(43,66)(44,68)(45,67)(51,53)(56,69)(62,71)(63,72)(64,70)

Polycyclic presentation:

2^9|b^2=f,c^2=i,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,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409046);