#9210501

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

4449 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)(25,26)(27,36)(28,33)(29,38)(30,34)(31,35)(32,37)(39,40);(3,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,24)(20,23)(21,22)(25,27)(26,32)(28,30)(29,34)(31,37)(33,39)(35,36)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,22)(21,24)(25,28,31,38)(26,33,35,29)(27,30,37,40)(32,39,36,34);(9,11)(10,15)(12,20)(13,19)(14,17)(16,23)(18,22)(21,24)(25,29)(26,28)(27,34)(30,32)(31,33)(35,38)(36,40)(37,39);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,21)(13,17)(15,19)(16,22)(18,24)(20,23)(25,26)(27,32)(28,33)(29,38)(30,39)(31,35)(34,40)(36,37);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,21)(19,20)(25,30,31,40)(26,34,35,39)(27,28,37,38)(29,36,33,32)

Polycyclic presentation:

2^9|a^2=h,c^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, 9117697);