#501853

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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	#33527	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

651 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)(12,28)(13,24)(14,20)(15,36)(16,21)(17,25)(18,29)(19,34)(23,31)(26,37)(27,30)(32,38)(33,39)(35,40);(1,3,4,7)(2,5,6,8)(9,11,16,28)(10,17,21,32)(12,24,22,40)(13,31,35,18)(14,27,15,26)(19,37,39,30)(20,23,36,29)(25,33,38,34);(2,6)(5,8)(9,10)(11,23)(12,18)(13,19)(14,36)(15,20)(16,21)(17,27)(22,31)(24,34)(25,30)(26,32)(28,29)(33,40)(35,39)(37,38);(9,12,16,22)(10,18,21,31)(11,24,28,40)(13,32,35,17)(14,30,15,37)(19,26,39,27)(20,38,36,25)(23,34,29,33)

Polycyclic presentation:

2^9|b^2=fi,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,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, 409049);