#9224299

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

4474 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)(3,8)(4,6)(5,7)(10,11)(13,21)(15,17)(19,22)(25,26,31,33)(27,28,35,29)(30,32,39,40)(34,36,38,37);(1,3,5,4)(2,6,7,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,24,22,23)(25,27)(26,28)(29,33)(30,38)(31,35)(32,37)(34,39)(36,40);(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,23,22,24)(25,28,31,29)(26,27,33,35)(30,36,39,37)(32,34,40,38);(1,2,5,7)(3,6,4,8)(9,11,14,10)(12,15,20,17)(13,16,21,18)(19,24,22,23)(25,29,31,28)(26,27,33,35)(30,36,39,37)(32,38,40,34);(1,4,5,3)(2,6,7,8)(9,12,14,20)(10,15,11,17)(13,22,21,19)(16,24,18,23)(25,30)(26,32)(27,34)(28,36)(29,37)(31,39)(33,40)(35,38);(1,2)(3,6)(4,8)(5,7)(9,13)(10,16)(11,18)(12,19)(14,21)(15,23)(17,24)(20,22)(25,27)(26,29)(28,33)(30,38)(31,35)(32,36)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9131495);