#6442668

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|c^2=g,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349864);