#8662636

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

4419 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8569832);