#6630357

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	#3006	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2064 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);(9,11,13,24,15,22,14,25)(10,16,18,21,20,30,19,32)(12,23,27,38,28,36,29,39)(17,31,26,35,33,37,34,40);(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)

Polycyclic presentation:

2^9|a^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,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, 6537553);