#504472

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	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 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)(9,10,17,34,14,22,26,18,16,24,13,21,15,23,32,35)(11,19,30,36,33,39,12,29,28,38,27,37,25,20,31,40);(10,18)(11,25)(13,26)(14,15)(17,32)(19,36)(20,29)(21,23)(22,34)(24,35)(27,30)(28,33)(37,38)(39,40);(2,5)(6,8)(9,11)(10,19)(12,26)(13,27)(14,33)(15,25)(16,28)(17,30)(18,29)(20,23)(21,37)(22,39)(24,38)(31,32)(34,36)(35,40);(1,3,4,7)(2,6,5,8)(9,12)(10,20)(11,26)(13,25)(14,30)(15,27)(16,31)(17,33)(18,29)(19,23)(21,36)(22,38)(24,39)(28,32)(34,37)(35,40)

Polycyclic presentation:

2^9|a^2=ei,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, 411668);