#467488

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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 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)(3,6)(4,5)(7,8)(9,10,13,18,14,19,26,16)(11,15,22,29,23,27,20,30)(12,17,25,32,24,33,38,31)(21,35,36,40,37,28,34,39);(9,11,13,22,14,23,26,20)(10,15,18,29,19,27,16,30)(12,21,25,36,24,37,38,34)(17,28,32,39,33,35,31,40);(2,5)(6,8)(10,16)(11,20)(12,24)(13,26)(15,27)(17,32)(18,19)(21,36)(22,23)(31,33)(34,37)(39,40);(1,3,4,7)(2,6,5,8)(9,12,13,25,14,24,26,38)(10,17,18,32,19,33,16,31)(11,21,22,36,23,37,20,34)(15,28,29,39,27,35,30,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 374684);