#8754833

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

4234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8662029);