#8720799

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

4425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627995);