#8429289

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

4361 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8336485);