#8399472

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

4462 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)(9,10,14,12)(11,17,18,15)(13,16,21,20)(19,23,24,22)(25,26)(27,32)(28,34)(29,30)(31,33)(35,40)(36,39)(37,38);(1,3,4,7)(2,5,6,8)(10,12)(11,18)(16,20)(19,24)(25,27,31,36)(26,32,33,39)(28,35,38,30)(29,34,40,37);(1,2)(3,5)(4,6)(7,8)(13,21)(16,20)(19,24)(22,23)(26,33)(27,36)(30,35)(34,37);(1,3)(2,5)(4,7)(6,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,28)(26,34)(27,35)(29,39)(30,36)(31,38)(32,40)(33,37);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,29)(26,35)(27,37)(28,32)(30,33)(31,40)(34,36)(38,39);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,30,31,35)(26,29,33,40)(27,28,36,38)(32,34,39,37)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306668);