#6629369

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	#32890	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^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, 6536565);