#509426

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

492 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:

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

Polycyclic presentation:

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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416622);