#9210505

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

4441 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=g,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, 9117701);