#6470830

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

1978 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)(3,5)(4,6)(7,8)(10,16)(11,20)(13,14)(17,32)(18,31)(19,28)(21,30)(22,35)(25,34)(26,29)(33,40)(36,39);(1,3)(2,5)(4,7)(6,8)(9,10,14,16,15,19,13,28)(11,21,24,32,20,17,23,30)(12,25,26,34,27,18,29,31)(22,33,37,39,35,36,38,40);(3,7)(5,8)(9,11,14,24,15,20,13,23)(10,17,16,30,19,21,28,32)(12,22,26,37,27,35,29,38)(18,33,31,39,25,36,34,40);(3,7)(5,8)(9,12)(10,18)(11,22)(13,29)(14,26)(15,27)(16,31)(17,33)(19,25)(20,35)(21,36)(23,38)(24,37)(28,34)(30,39)(32,40);(2,6)(5,8)(9,13,15,14)(10,16,19,28)(11,23,20,24)(12,26,27,29)(17,30,21,32)(18,34,25,31)(22,37,35,38)(33,40,36,39)

Polycyclic presentation:

2^9|b^2=f,c^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,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, 6378026);