#8069521

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

4361 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)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(26,32)(27,28)(34,39)(35,36);(1,2)(3,5)(4,6)(7,8)(25,26)(27,29)(28,33)(30,34)(31,32)(35,37)(36,40)(38,39);(10,14)(15,21)(16,22)(23,24)(25,27,31,28)(26,33,32,29)(30,36,38,35)(34,37,39,40);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,28,31,27)(26,33,32,29)(30,35,38,36)(34,37,39,40);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29,31,33)(26,27,32,28)(30,37,38,40)(34,35,39,36);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976717);