#8726703

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

4338 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633899);