#9222137

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

4641 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,e^2=g,f^2=h,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, 9129333);