#9040956

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

4449 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,e^2=gh,f^2=h,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, 8948152);