#9015562

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

4493 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=hi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922758);