#8393385

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

4406 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300581);