#9040997

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

4528 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)(3,7)(4,8)(5,6)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26,31,35)(27,32,36,30)(28,33,37,40)(29,34,38,39);(1,2,5,6)(3,4,8,7)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,27,31,36)(26,32,35,30)(28,38,37,29)(33,39,40,34);(3,8)(4,7)(10,12)(13,21)(15,17)(18,24)(27,36)(29,38)(30,32)(34,39);(2,6)(3,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28)(26,33)(27,29)(30,39)(31,37)(32,34)(35,40)(36,38);(1,3,5,8)(2,7,6,4)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,29)(26,34)(27,37)(28,36)(30,33)(31,38)(32,40)(35,39);(1,4,5,7)(2,8,6,3)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,27)(28,39)(29,33)(31,32)(34,37)(35,36)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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, 8948193);