#504378

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

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,22)(12,27)(13,17)(14,20)(15,21)(16,32)(18,26)(19,24)(23,38)(25,40)(28,39)(29,37)(30,35)(31,36)(33,34);(10,17)(11,23)(13,24)(15,16)(18,27)(19,21)(20,35)(22,39)(25,33)(26,29)(30,31)(32,36)(34,37)(38,40);(2,5)(6,8)(9,11,14,26)(10,18,20,22)(12,24,28,31)(13,25,30,33)(15,29,16,23)(17,34,35,40)(19,27,36,39)(21,38,32,37);(1,3,4,7)(2,6,5,8)(9,12,14,28)(10,18,20,22)(11,24,26,31)(13,29,30,23)(15,25,16,33)(17,27,35,39)(19,34,36,40)(21,37,32,38)

Polycyclic presentation:

2^9|c^2=i,d^2=f,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, 411574);