#6470823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470818
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 ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1948 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)(3,5)(4,6)(7,8)(10,16)(11,21)(13,14)(17,32)(18,31)(19,20)(22,30)(23,35)(26,34)(27,29)(33,40)(36,39);(1,3)(2,5)(4,7)(6,8)(9,10,14,16,15,20,13,19)(11,22,25,32,21,17,24,30)(12,26,27,34,28,18,29,31)(23,33,37,39,35,36,38,40);(3,7)(5,8)(9,11,14,25,15,21,13,24)(10,17,16,30,20,22,19,32)(12,23,27,37,28,35,29,38)(18,33,31,39,26,36,34,40);(3,7)(5,8)(9,12,15,28)(10,18,20,26)(11,23,21,35)(13,29,14,27)(16,31,19,34)(17,33,22,36)(24,38,25,37)(30,39,32,40);(2,6)(5,8)(9,13,15,14)(10,19,20,16)(11,24,21,25)(12,27,28,29)(17,32,22,30)(18,31,26,34)(23,37,35,38)(33,39,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6378019);