#8215477

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8122673);