#6686986

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=fi,e^2=f,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,h^c=hi,e^d=ei,h^e=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, 6594182);