#509772

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
prime radical	#33527	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

678 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)(9,10,13,19,15,20,30,14)(11,16,22,32,24,33,23,18)(12,25,27,34,28,35,29,17)(21,26,36,39,38,40,37,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,22)(14,18)(15,24)(17,21)(19,32)(20,33)(23,30)(25,36)(27,39)(28,40)(29,31)(34,38)(35,37);(9,12,15,28)(10,17,20,34)(11,21,24,38)(13,29,30,27)(14,25,19,35)(16,31,33,39)(18,26,32,40)(22,37,23,36);(2,6)(3,8)(9,11)(10,18)(12,21)(13,23)(14,16)(15,24)(17,26)(19,33)(20,32)(22,30)(25,31)(27,37)(28,38)(29,36)(34,40)(35,39)

Polycyclic presentation:

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

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, 416968);