#9161205

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

4555 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9068401);