#8720539

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

4533 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)(4,5)(6,8)(9,10)(11,16)(12,13)(14,18)(15,19)(17,22)(20,23)(21,24)(25,26)(27,34)(28,33)(29,32)(30,35)(31,36)(37,40)(38,39);(2,5)(6,7)(11,19)(15,16)(20,24)(21,23)(25,27,31,29)(26,32,36,34)(28,30,37,38)(33,39,40,35);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,15,18,16)(12,20,22,24)(13,21,17,23)(25,28,31,37)(26,33,36,40)(27,30,29,38)(32,39,34,35);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,20)(13,23)(14,19)(15,18)(17,21)(22,24)(25,29,31,27)(26,34,36,32)(28,30,37,38)(33,39,40,35);(9,12,14,22)(10,17,18,13)(11,20,19,24)(15,23,16,21)(25,26)(27,32)(28,33)(29,34)(30,39)(31,36)(35,38)(37,40);(9,13,14,17)(10,12,18,22)(11,21,19,23)(15,20,16,24)(25,30)(26,35)(27,37)(28,29)(31,38)(32,33)(34,40)(36,39)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627735);