#9040806

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

4454 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)(3,7)(4,8)(5,6)(25,26,30,33)(27,34,28,31)(29,32,37,40)(35,39,36,38);(1,2,5,6)(3,4,8,7)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(27,28)(31,34)(35,36)(38,39);(3,8)(4,7)(10,14)(12,21)(15,17)(18,22)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,39,40,38);(2,6)(3,8)(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,28,30,27)(26,31,33,34)(29,35,37,36)(32,39,40,38);(1,3,5,8)(2,7,6,4)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29,30,37)(26,32,33,40)(27,35,28,36)(31,38,34,39);(1,4,5,7)(2,8,6,3)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=gh,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, 8948002);