#8581789

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

4746 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)(3,4)(10,14)(15,21)(16,19)(22,24)(25,26,29,31)(27,33,35,30)(28,32,37,39)(34,40,36,38);(1,2,6,5)(3,8,4,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27,29,35)(26,30,31,33)(28,34,37,36)(32,38,39,40);(1,3,6,4)(2,7,5,8)(11,17)(15,21)(18,23)(22,24)(26,31)(30,33)(32,39)(38,40);(1,4,6,3)(2,8,5,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,37)(38,39);(1,5,6,2)(3,8,4,7)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,26)(27,30)(28,32)(29,31)(33,35)(34,38)(36,40)(37,39);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,29,37)(26,32,31,39)(27,34,35,36)(30,38,33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=i,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8488985);