#509822

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

662 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)(3,7)(4,6)(5,8)(9,10,18,26,16,22,36,38,14,12,15,21,17,23,35,34)(11,24,20,13,30,25,40,37,27,19,28,31,29,32,39,33);(1,3)(2,5)(4,8)(6,7)(9,11)(10,19)(12,24)(13,21)(14,27)(15,28)(16,30)(17,29)(18,20)(22,32)(23,25)(26,31)(33,38)(34,37)(35,39)(36,40);(9,12,14,10)(11,25,27,32)(13,28,31,20)(15,34,18,38)(16,22,17,23)(19,30,24,29)(21,36,26,35)(33,39,37,40);(2,6)(3,8)(9,13)(10,20)(11,26)(12,28)(14,31)(15,19)(16,33)(17,37)(18,24)(21,27)(22,39)(23,40)(25,35)(29,38)(30,34)(32,36)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417018);