#6492419

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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1777 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=hi,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, 6399615);