#8126989

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

4479 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=h,e^2=gh,f^2=g,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, 8034185);