#8214615

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

4536 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8121811);