#509502

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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	#33720	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

627 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)(11,22)(12,20)(13,19)(14,29)(15,21)(16,31)(17,24)(18,35)(23,37)(25,32)(26,38)(27,39)(28,36)(30,34)(33,40);(1,3,5,2)(4,7,8,6)(9,11,13,24)(10,17,19,22)(12,26,27,40)(14,23,28,35)(15,30,16,25)(18,36,37,29)(20,33,39,38)(21,32,31,34);(1,4,5,8)(2,6,3,7)(9,12)(11,23)(13,27)(14,16)(15,28)(17,32)(20,29)(21,31)(22,34)(24,35)(25,40)(26,30)(33,38)(36,39);(2,3)(6,7)(9,11)(10,18)(12,23)(13,24)(14,26)(15,25)(16,30)(17,29)(19,37)(20,32)(21,33)(22,36)(27,35)(28,40)(31,38)(34,39)

Polycyclic presentation:

2^9|b^2=e,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, 416698);