#8339991

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

4272 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)(3,4)(9,10,14,15)(11,19,20,16)(12,21,23,17)(13,18,24,22)(26,30)(31,36)(32,37)(38,40);(1,2,6,5)(3,8,4,7)(10,15)(11,20)(17,21)(18,22)(27,33)(31,36)(34,39)(38,40);(1,3,6,4)(2,7,5,8)(10,15)(13,24)(16,19)(17,21)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4,6,3)(2,8,5,7)(9,11)(10,16)(12,22)(13,17)(14,20)(15,19)(18,23)(21,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,5,6,2)(3,8,4,7)(9,12,14,23)(10,17,15,21)(11,18,20,22)(13,16,24,19)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,13,14,24)(10,18,15,22)(11,17,20,21)(12,16,23,19)(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=i,d^2=ghi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8247187);