#8695266

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

4411 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)(4,6)(5,8)(10,14)(11,17)(16,19)(18,20)(25,26)(27,30)(28,34)(29,38)(31,35)(32,36)(33,37)(39,40);(1,2,4,6)(3,5,8,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,27,32,37)(26,33,36,30)(28,31,29,39)(34,40,38,35);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,28,32,29)(26,34,36,38)(27,31,37,39)(30,35,33,40);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,29,32,28)(26,34,36,38)(27,31,37,39)(30,40,33,35);(9,11,13,17)(10,15,14,22)(12,20,21,18)(16,24,19,23)(25,30)(26,27)(28,35)(29,40)(31,34)(32,33)(36,37)(38,39);(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,31)(26,35)(27,29)(28,37)(30,38)(32,39)(33,34)(36,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8602462);