#9206603

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

4302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9113799);