#6807033

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1924 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,27)(13,19)(14,20)(15,21)(16,22)(18,28)(23,38)(24,34)(25,35)(26,37)(29,39)(30,40)(31,33)(32,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,14)(13,24)(15,32)(16,26)(18,38)(19,34)(20,27)(21,36)(22,37)(23,28)(25,30)(29,31)(33,39)(35,40);(9,12,15,28,16,29,13,25)(10,18,21,39,22,35,19,27)(11,23,24,14,26,30,32,31)(17,33,34,38,37,20,36,40);(2,6)(5,7)(9,13,16,15)(10,19,22,21)(11,24,26,32)(12,25,29,28)(14,30,31,23)(17,34,37,36)(18,27,35,39)(20,40,33,38);(3,8)(5,7)(9,14,15,30,16,31,13,23)(10,20,21,40,22,33,19,38)(11,25,24,12,26,28,32,29)(17,35,34,27,37,18,36,39)

Polycyclic presentation:

2^9|c^2=h,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,g^c=gi,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, 6714229);