#8429564

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

4643 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8336760);