#8417456

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

4586 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:

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

Polycyclic presentation:

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

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, 8324652);