#6816447

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1680 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)(11,15)(12,17)(13,16)(14,19)(18,26)(20,31)(21,29)(22,30)(23,28)(24,33)(25,34)(27,32)(35,38)(36,40)(37,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,24)(13,20)(14,23)(16,31)(17,33)(18,30)(19,28)(21,27)(22,26)(25,36)(29,32)(34,40)(35,39)(37,38);(2,6)(3,8)(10,16,19,18)(11,20,23,22)(15,28)(17,32,34,35)(21,36,37,24)(29,39)(30,31)(33,40);(2,6)(5,7)(12,25)(17,34)(21,37)(24,36)(27,38)(29,39)(32,35)(33,40);(9,12,26,38,14,25,13,27)(10,17,18,35,19,34,16,32)(11,21,20,36,23,37,22,24)(15,29,31,40,28,39,30,33)

Polycyclic presentation:

2^9|b^2=f,c^2=ghi,e^2=fhi,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, 6723643);