#8718910

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

4364 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=gi,f^2=gh,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, 8626106);