#6814818

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814815
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

1761 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)(9,10,14,19)(11,22,24,17)(12,25,27,38)(13,20,32,33)(15,31,16,21)(18,30,35,29)(23,28,37,40)(26,36,39,34);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,26)(13,31)(14,24)(15,20)(16,33)(18,28)(19,22)(21,32)(23,30)(25,34)(27,39)(29,37)(35,40)(36,38);(9,12,14,27)(10,18)(11,23)(13,28,32,40)(15,29,16,30)(17,34,22,36)(19,35)(20,26)(21,25)(24,37)(31,38)(33,39);(2,6)(5,7)(10,19)(12,27)(13,32)(17,22)(21,31)(23,37)(26,39)(29,30);(3,8)(5,7)(9,13,16,22,14,32,15,17)(10,20,21,24,19,33,31,11)(12,28,30,36,27,40,29,34)(18,26,25,37,35,39,38,23)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,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,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, 6722014);