#8114333

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

4204 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)(4,5)(6,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(26,28)(30,38)(32,35)(34,37);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(25,26,31,28)(27,32,29,35)(30,33,38,36)(34,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,31,29)(26,32,28,35)(30,34,38,37)(33,39,36,40);(1,3,4,8)(2,6,5,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,26)(27,32,29,35)(30,36,38,33)(34,39,37,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,31,27)(26,32,28,35)(30,37,38,34)(33,39,36,40);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,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, 8021529);