#6467860

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 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)(3,6)(4,5)(7,8)(9,10,16,20)(11,15,19,14)(12,17,26,33)(13,18,28,35)(21,25,32,24)(22,30,36,29)(23,31,27,34)(37,40,38,39);(9,11,14,10,16,19,15,20)(12,17,24,32,26,33,25,21)(13,18,29,36,28,35,30,22)(23,38,39,34,27,37,40,31);(2,5)(6,8)(9,12,14,24,16,26,15,25)(10,17,11,21,20,33,19,32)(13,27,29,40,28,23,30,39)(18,34,22,38,35,31,36,37);(9,13,14,29,16,28,15,30)(10,18,11,22,20,35,19,36)(12,23,24,39,26,27,25,40)(17,31,21,37,33,34,32,38);(1,3,4,7)(2,6,5,8)(13,28)(18,35)(22,36)(23,27)(29,30)(31,34)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375056);