#8700506

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

4529 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)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(25,26)(27,34)(28,36)(29,31)(30,33)(32,37)(35,39)(38,40);(3,7)(5,8)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,39,36,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,11)(10,16,18,15)(13,19,22,20)(17,23,21,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,37)(38,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,28)(26,32)(27,35)(29,38)(30,37)(31,39)(33,36)(34,40);(9,13,14,22)(10,17,18,21)(11,19,12,20)(15,23,16,24)(25,29,30,27)(26,31,33,34)(28,35,37,38)(32,40,36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607702);