#9164115

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

4535 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)(11,19)(12,17)(13,21)(14,15)(16,20)(18,24)(22,23)(25,26)(27,32)(28,36)(29,33)(30,34)(31,37)(35,39)(38,40);(1,2)(3,8)(4,6)(5,7)(10,15)(12,23)(16,19)(18,21)(25,27)(26,29)(28,35)(30,33)(31,39)(32,34)(36,40)(37,38);(1,3,5,4)(2,6,7,8)(10,15)(11,20)(17,22)(18,21)(28,37)(31,36)(35,38)(39,40);(3,4)(6,8)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,28)(26,31)(27,35)(29,39)(30,37)(32,40)(33,38)(34,36);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,27)(26,32)(28,38)(29,34)(30,33)(31,39)(35,37)(36,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,29,30,32)(26,33,34,27)(28,39,37,40)(31,38,36,35)

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=i,f^2=g,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, 9071311);