#6630363

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
prime radical	#3006	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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)(13,26)(15,27)(16,29)(18,32)(20,33)(22,35)(24,25)(30,31)(34,37)(39,40);(2,5)(6,8)(9,10)(11,20)(12,16)(13,18)(14,17)(15,22)(19,33)(21,36)(23,29)(24,31)(25,30)(26,32)(27,35)(28,38)(34,40)(37,39);(9,11,13,19,14,22,26,35)(10,15,18,27,17,20,32,33)(12,21,24,34,25,38,23,37)(16,28,31,39,30,36,29,40);(9,12)(10,16)(11,21)(13,24)(14,25)(15,28)(17,30)(18,31)(19,34)(20,36)(22,38)(23,26)(27,39)(29,32)(33,40)(35,37);(1,3,4,7)(2,6,5,8)(10,17)(15,20)(16,30)(18,32)(27,33)(28,36)(29,31)(39,40)

Polycyclic presentation:

2^9|c^2=g,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^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, 6537559);