#6629371

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628867
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3989 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)(9,10,14,18)(11,19,22,33)(12,16,23,31)(13,25,28,17)(15,26,30,21)(20,36,37,35)(24,34,27,32)(29,39,40,38);(2,5)(6,8);(9,11)(10,15)(12,20)(13,26)(14,22)(16,29)(17,33)(18,30)(19,25)(21,28)(23,37)(24,39)(27,38)(31,40)(32,35)(34,36);(9,12,14,23)(10,16,18,31)(11,20,22,37)(13,27,28,24)(15,29,30,40)(17,34,25,32)(19,35,33,36)(21,39,26,38);(1,3,4,7)(2,6,5,8)(12,23)(16,31)(20,37)(24,27)(29,40)(32,34)(35,36)(38,39)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,h^c=hi,e^d=ei,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, 6536567);