#6629385

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3920 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)(9,10,14,13)(11,19,21,16)(12,23,25,32)(15,27,28,18)(17,29,24,26)(20,35,36,40)(22,31,38,34)(30,37,33,39);(2,5)(6,8)(9,11)(10,16)(12,20)(13,19)(14,21)(15,22)(17,30)(18,31)(23,40)(24,33)(25,36)(26,37)(27,34)(28,38)(29,39)(32,35);(9,12)(10,17)(11,20)(13,24)(14,25)(15,29)(16,30)(18,32)(19,33)(21,36)(22,39)(23,27)(26,28)(31,35)(34,40)(37,38);(9,13,14,10)(11,19,21,16)(12,24,25,17)(15,18,28,27)(20,33,36,30)(22,31,38,34)(23,29,32,26)(35,37,40,39);(1,3,4,7)(2,6,5,8)(11,21)(16,19)(20,36)(22,38)(30,33)(31,34)(35,40)(37,39)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,g^c=gi,h^c=hi,g^d=gi

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, 6536581);