#6573811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4581 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)(3,4)(9,10,14,17)(11,19,22,32)(12,16,23,30)(13,25,27,18)(15,26,29,21)(20,33,34,40)(24,37,36,31)(28,38,39,35);(5,6)(7,8)(9,11,14,22)(10,15,17,29)(12,20,23,34)(13,26,27,21)(16,28,30,39)(18,32,25,19)(24,38,36,35)(31,40,37,33);(2,4)(6,8)(9,12,14,23)(10,16,17,30)(11,20,22,34)(13,24,27,36)(15,28,29,39)(18,31,25,37)(19,33,32,40)(21,35,26,38);(5,6)(7,8)(12,23)(16,30)(20,34)(24,36)(28,39)(31,37)(33,40)(35,38);(6,8)(10,17)(15,29)(16,30)(18,25)(19,32)(28,39)(31,37)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481007);