#9222185

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

4497 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)(9,10,14,15)(11,19,21,16)(12,17,22,20)(13,23,24,18)(26,32)(28,38)(33,35)(36,39);(1,2,5,8)(3,4,7,6)(10,15)(12,22)(16,19)(18,23)(25,26)(27,33)(28,34)(29,35)(30,36)(31,32)(37,38)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,18)(13,20)(14,21)(15,19)(17,24)(22,23)(25,27,31,29)(26,33,32,35)(28,36,38,39)(30,37,40,34);(3,7)(4,6)(9,11)(10,16)(12,23)(13,17)(14,21)(15,19)(18,22)(20,24)(25,28)(26,34)(27,36)(29,39)(30,33)(31,38)(32,37)(35,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,17,15,20)(11,18,21,23)(13,16,24,19)(25,29,31,27)(26,35,32,33)(28,36,38,39)(30,37,40,34);(1,4,5,6)(2,7,8,3)(9,13,14,24)(10,18,15,23)(11,20,21,17)(12,19,22,16)(25,30)(26,36)(27,37)(28,35)(29,34)(31,40)(32,39)(33,38)

Polycyclic presentation:

2^9|a^2=h,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, 9129381);