#8550906

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

4382 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)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,2,4,6)(3,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(26,30)(27,34)(32,36)(35,40);(1,2)(3,5)(4,6)(7,8)(11,17)(14,21)(18,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=h,e^2=h,f^2=gi,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, 8458102);