#9184105

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

4737 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)(9,10,14,13)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26,30,34)(27,31,37,32)(28,29,33,39)(35,36,38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,24)(21,23)(25,27)(26,31)(28,35)(29,36)(30,37)(32,34)(33,38)(39,40);(1,2)(3,5)(4,6)(7,8)(12,20)(16,21)(18,24)(22,23)(25,27,30,37)(26,31,34,32)(28,35,33,38)(29,36,39,40);(9,12,14,20)(10,16,13,21)(11,18,19,24)(15,22,17,23)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,35)(39,40);(1,3)(2,5)(4,7)(6,8)(9,13,14,10)(11,17,19,15)(12,16,20,21)(18,22,24,23)(25,28)(26,29)(27,35)(30,33)(31,36)(32,40)(34,39)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,24)(21,22)(25,29)(26,33)(27,36)(28,34)(30,39)(31,38)(32,35)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=g,e^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, 9091301);