#6630361

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629034
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

2080 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:

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

Polycyclic presentation:

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

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, 6537557);