#6629710

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2175 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)(3,5)(4,6)(7,8)(9,10,14,16)(11,19,22,32)(12,24,26,17)(13,27,20,18)(15,28,29,23)(21,35,36,33)(25,31,34,39)(30,37,38,40);(1,3,4,7)(2,5,6,8)(9,11,20,28,14,22,13,23)(10,15,27,32,16,29,18,19)(12,21,34,40,26,36,25,37)(17,30,39,35,24,38,31,33);(2,6)(5,8)(10,16)(15,29)(17,24)(18,27)(19,32)(30,38)(31,39)(33,35);(9,12,13,25,14,26,20,34)(10,17,18,31,16,24,27,39)(11,21,23,37,22,36,28,40)(15,30,19,33,29,38,32,35);(11,22)(15,29)(19,32)(21,36)(23,28)(30,38)(33,35)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,e^b=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, 6536906);