#8583459

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

4584 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)(10,11)(15,17)(16,18)(22,23)(25,26,32,36)(27,37,38,33)(28,35,29,34)(30,39,40,31);(1,2,4,6)(3,5,8,7)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(25,27,32,38)(26,33,36,37)(28,31,29,39)(30,35,40,34);(1,2)(3,5)(4,6)(7,8)(12,19)(15,17)(20,24)(22,23)(25,28,32,29)(26,34,36,35)(27,31,38,39)(30,37,40,33);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,29,32,28)(26,35,36,34)(27,31,38,39)(30,37,40,33);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,30)(26,31)(27,34)(28,33)(29,37)(32,40)(35,38)(36,39);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,31)(26,30)(27,29)(28,38)(32,39)(33,35)(34,37)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8490655);