#6514386

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
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

1768 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=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, 6421582);