#8720831

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

4413 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)(3,7)(4,6)(5,8)(9,10,15,21)(11,16,23,24)(12,18,22,17)(13,14,20,19)(26,30)(31,36)(32,37)(38,40);(1,2,4,6)(3,5,8,7)(9,11,15,23)(10,16,21,24)(12,14,22,19)(13,18,20,17)(27,33)(31,36)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12,15,22)(10,17,21,18)(11,19,23,14)(13,16,20,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,19)(13,16)(14,23)(15,22)(17,21)(20,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,13)(10,19)(11,18)(12,16)(14,21)(15,20)(17,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,14,15,19)(10,20,21,13)(11,22,23,12)(16,17,24,18)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8628027);