#8533509

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

4489 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440705);