#9436243

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9343439);