#6456447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
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

1840 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)(9,10,13,18,14,19,26,15)(11,20,23,27,21,16,29,30)(12,17,25,32,24,33,37,28)(22,34,36,38,35,31,39,40);(10,15)(11,21)(12,24)(13,26)(16,30)(17,32)(18,19)(20,27)(22,35)(28,33)(31,40)(34,38);(2,5)(6,8)(9,11,13,23,14,21,26,29)(10,16,18,30,19,20,15,27)(12,22,25,36,24,35,37,39)(17,31,32,40,33,34,28,38);(9,12,13,25,14,24,26,37)(10,17,18,32,19,33,15,28)(11,22,23,36,21,35,29,39)(16,31,30,40,20,34,27,38);(1,3,4,7)(2,6,5,8)(11,21)(16,20)(22,35)(23,29)(27,30)(31,34)(36,39)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363643);