#504883

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
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

594 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)(3,7)(4,6)(5,8)(9,10)(11,23)(12,28)(13,32)(14,20)(15,21)(16,19)(17,26)(18,24)(22,33)(25,31)(27,37)(29,35)(30,40)(34,36)(38,39);(1,3,4,8)(2,5,6,7)(9,10,16,32,15,22,34,29,14,20,35,36,33,21,13,19)(11,24,28,12,18,23,38,31,26,37,40,30,27,17,25,39);(2,6)(5,7)(9,11)(10,17)(12,29)(13,25)(14,26)(15,18)(16,28)(19,30)(20,23)(21,37)(22,24)(27,33)(31,36)(32,39)(34,38)(35,40);(2,6)(5,7)(9,12,16,23,15,31,34,37,14,30,35,17,33,39,13,24)(10,18,32,38,22,26,29,40,20,27,36,25,21,11,19,28)

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,d^c=di,g^c=gi

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, 412079);