#9028834

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) = 128

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=gh,e^2=gh,f^2=i,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, 8936030);