#504470

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	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

577 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)(3,6)(4,5)(7,8)(9,10,13,20,14,21,36,53,16,23,38,54,15,22,37,46)(11,24,27,42,28,57,64,44,30,58,65,43,29,52,34,17)(12,18,31,47,39,55,67,51,33,49,66,50,32,48,35,19)(25,56,59,71,62,70,72,68,60,69,63,45,26,41,61,40);(9,11,16,30)(10,17,23,44)(12,26,33,62)(13,34,38,64)(14,29,15,28)(18,45,49,71)(19,41,51,70)(20,52,54,57)(21,43,22,42)(24,53,58,46)(25,39,60,32)(27,36,65,37)(31,63,66,59)(35,61,67,72)(40,55,68,48)(47,69,50,56);(2,5)(6,8)(9,12)(10,18)(11,25)(13,31)(14,39)(15,32)(16,33)(17,40)(19,46)(20,47)(21,55)(22,48)(23,49)(24,56)(26,29)(27,59)(28,62)(30,60)(34,61)(35,37)(36,67)(38,66)(41,52)(42,71)(43,45)(44,68)(50,54)(51,53)(57,70)(58,69)(63,65)(64,72);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,26)(13,35)(14,32)(15,39)(16,33)(17,41)(18,46)(20,48)(21,50)(22,47)(23,51)(24,45)(25,29)(27,63)(28,60)(30,62)(31,37)(34,61)(36,66)(38,67)(40,52)(42,69)(43,56)(44,70)(49,53)(54,55)(57,68)(58,71)(59,65)(64,72)

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,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, 411666);