#8720731

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

4436 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)(3,7)(4,5)(6,8)(9,10,15,17)(11,19,14,16)(12,22,18,13)(20,24,23,21)(25,26)(27,35)(28,39)(29,40)(30,34)(31,32)(33,36)(37,38);(2,5)(6,7)(9,11,15,14)(10,16,17,19)(12,23,18,20)(13,21,22,24)(26,32)(27,36)(28,40)(34,38);(1,2)(3,6)(4,5)(7,8)(10,17)(12,18)(16,19)(20,23)(25,27,31,36)(26,33,32,35)(28,37,40,30)(29,38,39,34);(1,3)(2,6)(4,8)(5,7)(9,12,15,18)(10,13,17,22)(11,20,14,23)(16,24,19,21)(25,28)(26,29)(27,37)(30,36)(31,40)(32,39)(33,38)(34,35);(9,13)(10,18)(11,21)(12,17)(14,24)(15,22)(16,20)(19,23)(25,29)(26,28)(27,38)(30,35)(31,39)(32,40)(33,37)(34,36);(9,14,15,11)(10,16,17,19)(12,20,18,23)(13,21,22,24)(25,30,31,37)(26,34,32,38)(27,28,36,40)(29,35,39,33)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=gi,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, 8627927);