#6814808

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814805
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60773	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^d=ei

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, 6722004);