#8127124

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

4463 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034320);