#467489

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

806 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)(3,6)(4,5)(7,8)(9,10,14,19,15,20,13,18)(11,16,24,31,25,32,23,21)(12,17,29,34,27,26,28,33)(22,36,40,35,38,30,39,37);(9,11,14,24,15,25,13,23)(10,16,19,31,20,32,18,21)(12,22,29,40,27,38,28,39)(17,30,34,37,26,36,33,35);(2,5)(6,8)(9,10,15,20)(11,21,25,31)(12,26,27,17)(13,19,14,18)(16,24,32,23)(22,37,38,35)(28,33,29,34)(30,39,36,40);(1,3,4,7)(2,6,5,8)(9,12,13,28,15,27,14,29)(10,17,18,33,20,26,19,34)(11,22,23,39,25,38,24,40)(16,30,21,35,32,36,31,37)

Polycyclic presentation:

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

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, 374685);