#6573464

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4555 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)(5,6)(7,8)(9,10,14,15)(11,16,22,19)(12,17,24,28)(13,18,27,29)(20,30,34,33)(21,31,37,25)(23,32,38,26)(35,36,39,40);(2,4)(9,11,14,22)(10,16,15,19)(12,20,24,34)(13,25,27,31)(17,30,28,33)(18,21,29,37)(23,36,38,40)(26,35,32,39);(6,8)(10,15)(11,19)(12,17)(16,22)(18,29)(20,34)(21,25)(23,32)(24,28)(26,38)(31,37)(35,39);(9,12,14,24)(10,17,15,28)(11,20,22,34)(13,26,27,32)(16,30,19,33)(18,23,29,38)(21,36,37,40)(25,35,31,39);(9,13,14,27)(10,18,15,29)(11,21,22,37)(12,23,24,38)(16,31,19,25)(17,32,28,26)(20,35,34,39)(30,36,33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480660);