#6630358

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	#32945	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 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)(12,22)(13,25)(15,26)(16,29)(18,32)(21,34)(23,24)(28,38)(30,31)(33,35)(37,40);(2,5)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,26)(20,27)(21,28)(22,29)(23,31)(24,30)(25,32)(33,37)(34,38)(35,40)(36,39);(9,11,13,19,14,21,25,34)(10,15,18,26,17,28,32,38)(12,20,23,33,24,36,22,35)(16,27,31,37,30,39,29,40);(9,12)(10,16)(11,20)(13,23)(14,24)(15,27)(17,30)(18,31)(19,33)(21,36)(22,25)(26,37)(28,39)(29,32)(34,35)(38,40);(1,3,4,7)(2,6,5,8)(10,17)(15,28)(16,30)(18,32)(26,38)(27,39)(29,31)(37,40)

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,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, 6537554);