#417727

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2338 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)(3,4)(5,6)(7,8)(9,10,16,22)(11,23,27,36)(12,28,32,39)(13,33,24,19)(14,21,15,20)(17,26,37,25)(18,30,35,31)(29,40,34,38);(9,11,16,27)(10,17,22,37)(12,29,32,34)(13,30,24,31)(14,26,15,25)(18,38,35,40)(19,28,33,39)(20,36,21,23);(2,4)(9,12)(10,18)(11,24)(13,27)(14,30)(15,31)(16,32)(17,19)(20,39)(21,28)(22,35)(23,38)(25,29)(26,34)(33,37)(36,40);(6,8)(9,13,16,24)(10,19,22,33)(11,12,27,32)(14,29,15,34)(17,35,37,18)(20,40,21,38)(23,39,36,28)(25,31,26,30)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324923);