#6815112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814811
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

1883 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=f,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,d^c=di

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, 6722308);