#9222220

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
prime radical	#51142	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=g,e^2=g,f^2=gh,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, 9129416);