#9910718

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

4550 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817914);