#8429505

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4155 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=gh,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, 8336701);