#9166554

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

4447 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)(12,17)(13,22)(14,15)(18,24)(19,23)(20,21)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40);(3,7)(5,8)(10,15)(12,21)(16,23)(18,22)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,15)(11,19)(17,20)(18,22)(27,34)(30,31)(35,40)(38,39);(9,11,14,19)(10,16,15,23)(12,22,21,18)(13,20,24,17)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,21)(10,17,15,20)(11,18,19,22)(13,16,24,23)(25,26)(27,30)(28,36)(29,33)(31,34)(32,37)(35,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,24)(10,18,15,22)(11,20,19,17)(12,23,21,16)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40)

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9073750);