#6432096

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431723
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23406	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3295 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,18)(13,19)(14,28)(15,20)(17,25)(21,30)(22,31)(23,36)(24,32)(26,34)(27,35)(29,37)(33,39)(38,40);(2,5)(6,8)(9,11,15,24)(10,16,20,32)(12,21,26,38)(13,23,14,22)(17,33,35,29)(18,30,34,40)(19,36,28,31)(25,39,27,37);(9,12)(10,17)(11,21)(13,25)(14,27)(15,26)(16,29)(18,28)(19,34)(20,35)(22,37)(23,39)(24,38)(30,36)(31,40)(32,33);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,27)(14,25)(15,26)(16,30)(17,28)(19,35)(20,34)(22,39)(23,37)(24,38)(29,36)(31,33)(32,40);(9,13,15,14)(10,19,20,28)(11,22,24,23)(12,25,26,27)(16,31,32,36)(17,34,35,18)(21,37,38,39)(29,40,33,30)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339292);