#8429377

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

4581 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)(3,7)(9,10,14,18)(11,12,16,15)(13,17,22,21)(19,20,24,23)(25,26,31,36)(27,30,35,32)(28,34,29,33)(37,40,38,39);(1,2,4,5)(3,6,7,8)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,27,31,35)(26,32,36,30)(28,37,29,38)(33,40,34,39);(1,2)(3,6)(4,5)(7,8)(25,28,31,29)(26,33,36,34)(27,37,35,38)(30,39,32,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,29,31,28)(26,34,36,33)(27,38,35,37)(30,40,32,39);(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,30,31,32)(26,35,36,27)(28,39,29,40)(33,38,34,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,e^2=g,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, 8336573);