#509187

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416383);