#6630368

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	#32945	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2072 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)(11,16)(12,23)(13,27)(14,19)(15,20)(17,21)(18,22)(24,33)(25,38)(26,30)(28,31)(29,34)(32,39)(35,36)(37,40);(2,5)(6,8)(9,11,15,14)(10,16,20,19)(12,24,26,21)(13,22,28,29)(17,23,33,30)(18,31,34,27)(25,37,39,35)(32,36,38,40);(9,12,11,21,15,26,14,24)(10,17,19,23,20,33,16,30)(13,25,22,35,28,39,29,37)(18,32,27,36,34,38,31,40);(9,13)(10,18)(11,22)(12,25)(14,29)(15,28)(16,31)(17,32)(19,27)(20,34)(21,35)(23,36)(24,37)(26,39)(30,40)(33,38);(1,3,4,7)(2,6,5,8)(12,26)(17,33)(21,24)(23,30)(25,39)(32,38)(35,37)(36,40)

Polycyclic presentation:

2^9|b^2=i,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^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, 6537564);