#504468

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

615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,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, 411664);