#8393237

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

4546 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300433);