#9224331

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

4473 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)(4,7)(5,6)(9,10)(11,16)(12,22)(13,14)(15,18)(17,21)(19,24)(20,23)(25,26,30,35)(27,34,37,31)(28,38,39,32)(29,40,36,33);(1,3,5,7)(2,4,6,8)(9,11)(10,16)(12,23)(13,19)(14,24)(15,21)(17,18)(20,22)(25,27,30,37)(26,31,35,34)(28,36,39,29)(32,33,38,40);(9,12,15,13)(10,14,18,22)(11,19,21,23)(16,20,17,24)(25,28,30,39)(26,32,35,38)(27,36,37,29)(31,33,34,40);(2,6)(4,8)(9,13,15,12)(10,14,18,22)(11,19,21,23)(16,24,17,20)(25,29)(26,33)(27,28)(30,36)(31,38)(32,34)(35,40)(37,39);(1,2)(3,4)(5,6)(7,8)(9,14)(10,13)(11,20)(12,18)(15,22)(16,23)(17,19)(21,24)(25,27)(26,34)(28,36)(29,39)(30,37)(31,35)(32,40)(33,38);(1,4)(2,7)(3,6)(5,8)(9,11)(10,17)(12,19)(13,23)(14,24)(15,21)(16,18)(20,22)(25,26)(27,31)(28,32)(29,40)(30,35)(33,36)(34,37)(38,39)

Polycyclic presentation:

2^9|a^2=hi,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, 9131527);