#9040615

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

4578 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947811);