#6630370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629032
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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2056 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)(9,10,15,20)(11,21,22,32)(12,26,28,34)(13,19,14,18)(16,25,30,24)(17,29,33,27)(23,37,36,31)(35,39,40,38);(2,5)(6,8)(11,22)(16,30)(21,32)(23,36)(24,25)(31,37)(35,40)(38,39);(9,11,14,25,15,22,13,24)(10,16,19,32,20,30,18,21)(12,23,29,39,28,36,27,38)(17,31,34,40,33,37,26,35);(9,12)(10,17)(11,23)(13,27)(14,29)(15,28)(16,31)(18,26)(19,34)(20,33)(21,35)(22,36)(24,38)(25,39)(30,37)(32,40);(1,3,4,7)(2,6,5,8)(11,22)(16,30)(21,32)(23,36)(24,25)(31,37)(35,40)(38,39)

Polycyclic presentation:

2^9|a^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,c^b=ci,e^c=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, 6537566);