#8119663

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

4705 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,17)(13,21)(14,16)(18,20)(19,23)(22,24)(28,36)(32,37)(34,40)(38,39);(2,6)(3,7)(9,11)(10,15)(12,21)(13,17)(14,20)(16,18)(19,22)(23,24)(26,30)(31,33)(32,37)(38,39);(1,3,5,7)(2,4,6,8)(10,16)(11,20)(17,24)(19,21)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,17)(11,19)(13,18)(14,22)(15,23)(16,24)(20,21);(9,11)(10,18)(12,19)(13,17)(14,20)(15,16)(21,22)(23,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,13)(10,19)(11,17)(12,18)(14,23)(15,22)(16,21)(20,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|c^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8026859);