#509253

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

684 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=i,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416449);