#8630864

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

4437 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)(3,4)(5,8)(6,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(26,30)(27,33)(32,38)(35,36);(1,3)(2,4)(5,6)(7,8)(10,14)(15,17)(16,21)(22,23)(25,26,29,30)(27,34,33,31)(28,32,37,38)(35,40,36,39);(1,4)(2,5)(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26,29,30)(27,31,33,34)(28,32,37,38)(35,39,36,40);(1,3,6,5)(2,4,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,31)(28,36)(29,33)(30,34)(32,40)(35,37)(38,39);(1,5,6,3)(2,4,7,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,40)(37,38);(1,4)(2,3)(5,7)(6,8)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,28,29,37)(26,32,30,38)(27,35,33,36)(31,39,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538060);