#504879

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	#13307	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

572 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)(3,7)(4,6)(5,8)(9,10)(11,24)(12,32)(13,38)(14,42)(15,21)(16,22)(17,28)(18,27)(19,39)(20,25)(23,43)(26,50)(29,58)(30,59)(31,66)(33,52)(34,48)(35,69)(36,56)(37,46)(40,67)(41,71)(44,54)(45,72)(47,65)(49,70)(51,62)(53,63)(55,60)(57,64)(61,68);(1,3,4,8)(2,5,6,7)(9,11,25,60,16,31,44,65,15,29,63,34,43,30,14,28)(10,17,42,59,23,48,53,58,21,47,54,66,22,55,20,24)(12,33,61,72,37,39,64,13,27,62,70,41,36,26,35,40)(18,38,57,19,46,45,68,52,32,67,69,50,56,71,49,51);(2,6)(5,7)(9,12)(10,18)(11,26)(13,34)(14,35)(15,27)(16,37)(17,45)(19,24)(20,49)(21,32)(22,56)(23,46)(25,61)(28,41)(29,39)(30,62)(31,33)(36,43)(38,55)(40,60)(42,57)(44,64)(47,71)(48,67)(50,58)(51,66)(52,59)(53,68)(54,69)(63,70)(65,72);(2,6)(5,7)(9,13,25,62,16,41,44,26,15,40,63,33,43,72,14,39)(10,19,42,45,23,52,53,67,21,50,54,71,22,51,20,38)(11,27,60,70,31,36,65,35,29,12,34,61,30,37,28,64)(17,46,59,68,48,32,58,69,47,56,66,49,55,18,24,57)

Polycyclic presentation:

2^9|b^2=eh,d^2=efhi,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, 412075);