#8429702

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

4672 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8336898);