#6437716

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4578 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)(11,15)(12,16)(13,18)(14,17)(19,26)(20,30)(21,32)(22,31)(23,33)(24,35)(25,34)(27,29)(28,36)(37,39)(38,40);(2,4)(9,11,14,22)(10,15,17,31)(12,23,25,38)(13,26,29,21)(16,33,34,40)(18,19,27,32)(20,28,37,24)(30,36,39,35);(6,8)(9,10)(11,19)(12,16)(13,27)(14,17)(15,26)(18,29)(20,33)(21,31)(22,32)(23,30)(24,36)(25,34)(28,35)(37,40)(38,39);(9,12,14,25)(10,16,17,34)(11,20,22,37)(13,28,29,24)(15,30,31,39)(18,36,27,35)(19,33,32,40)(21,38,26,23);(10,17)(15,31)(16,34)(18,27)(19,32)(30,39)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,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, 6344912);