#8384260

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

4511 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gh,e^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, 8291456);