#432467

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

931 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,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, 339663);