#8119565

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

4279 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=i,f^2=g,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, 8026761);