#426946

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

828 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=f,d^2=e,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, 334142);