#9028762

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

4418 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=h,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, 8935958);