#9224288

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

4480 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,19,21,15)(12,16,22,20)(13,17,24,23)(25,26,31,33)(27,28,35,29)(30,32,39,40)(34,36,38,37);(1,2,4,6)(3,5,8,7)(9,11,14,21)(10,15,18,19)(12,17,22,23)(13,20,24,16)(25,27)(26,28)(29,33)(30,38)(31,35)(32,37)(34,39)(36,40);(1,2)(3,5)(4,6)(7,8)(25,28,31,29)(26,27,33,35)(30,36,39,37)(32,34,40,38);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,20)(13,23)(14,18)(16,22)(17,24)(19,21)(25,29,31,28)(26,27,33,35)(30,36,39,37)(32,38,40,34);(9,12)(10,16)(11,17)(13,15)(14,22)(18,20)(19,24)(21,23)(25,30)(26,32)(27,34)(28,36)(29,37)(31,39)(33,40)(35,38);(9,13)(10,17)(11,20)(12,19)(14,24)(15,22)(16,21)(18,23)(25,27)(26,29)(28,33)(30,38)(31,35)(32,36)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=i,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, 9131484);