#9221220

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

4494 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=h,f^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, 9128416);