#9041084

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

4743 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=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, 8948280);