#8492822

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

4529 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8400018);