#9236500

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

4637 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143696);