#8345983

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

4272 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)(3,8)(4,6)(5,7)(9,10,15,12)(11,16,14,18)(13,17,22,20)(19,23,21,24)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(1,2,6,4)(3,8,5,7)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(28,36)(32,38)(35,39)(37,40);(1,2,6,4)(3,7,5,8)(11,14)(16,18)(19,21)(23,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(1,3,6,5)(2,7,4,8)(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,36)(33,39)(34,40);(1,4,6,2)(3,7,5,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,5,6,3)(2,7,4,8)(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)

Polycyclic presentation:

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

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, 8253179);