#509978

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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

507 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)(9,10,13,18,31,34,40,39,15,20,33,38,14,19,32,30)(11,16,22,12,23,26,37,27,25,35,17,29,24,21,36,28);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,18)(13,22)(14,24)(15,25)(17,33)(19,21)(20,35)(23,31)(26,34)(27,39)(28,30)(29,38)(32,36)(37,40);(1,4,5,8)(2,6,3,7)(9,10)(11,21)(12,17)(13,30)(14,34)(15,20)(16,24)(18,32)(19,31)(22,29)(23,35)(25,26)(27,37)(28,36)(33,39)(38,40);(2,3)(6,7)(9,12,15,29)(10,17,20,22)(11,18,25,38)(13,16,33,35)(14,27,31,28)(19,36,34,37)(21,32,26,40)(23,30,24,39)

Polycyclic presentation:

2^9|a^2=f,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=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417174);