#509976

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
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
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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)(3,5)(4,6)(7,8)(9,10,18,26,16,22,34,37,14,12,15,21,17,23,33,32)(11,24,29,30,35,25,38,31,27,19,20,13,28,39,40,36);(1,3,5,2)(4,7,8,6)(9,11,14,27)(10,19,12,24)(13,21,30,26)(15,20,18,29)(16,35,17,28)(22,39,23,25)(31,37,36,32)(33,40,34,38);(1,4,5,8)(2,6,3,7)(9,12,14,10)(11,25,27,39)(13,29,30,20)(15,32,18,37)(16,22,17,23)(19,35,24,28)(21,34,26,33)(31,38,36,40);(2,3)(6,7)(9,13,14,30)(10,20,12,29)(11,26,27,21)(15,24,18,19)(16,31,17,36)(22,38,23,40)(25,33,39,34)(28,37,35,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=e,d^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, 417172);