#8269922

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

4519 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:

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

Polycyclic presentation:

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

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, 8177118);