#6573812

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
prime radical	C₂×#3489	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4610 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)(9,10,15,20)(11,21,24,34)(12,17,25,31)(13,19,14,18)(16,28,30,23)(22,35,36,40)(26,33,27,32)(29,38,39,37);(5,6)(7,8)(9,11,15,24)(10,16,20,30)(12,22,25,36)(13,23,14,28)(17,29,31,39)(18,21,19,34)(26,37,27,38)(32,35,33,40);(2,4)(6,8)(9,12,15,25)(10,17,20,31)(11,22,24,36)(13,26,14,27)(16,29,30,39)(18,32,19,33)(21,35,34,40)(23,37,28,38);(5,6)(7,8)(12,25)(17,31)(22,36)(26,27)(29,39)(32,33)(35,40)(37,38);(6,8)(9,13,15,14)(10,18,20,19)(11,23,24,28)(12,26,25,27)(16,21,30,34)(17,32,31,33)(22,37,36,38)(29,35,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^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, 6481008);