#9668361

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

4768 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)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26,31,29)(27,35,37,32)(28,38,39,33)(30,40,36,34);(2,5)(6,7)(10,14)(15,17)(16,20)(23,24)(25,27,31,37)(26,32,29,35)(28,36,39,30)(33,34,38,40);(1,2)(3,6)(4,5)(7,8)(10,14)(11,18)(16,20)(19,21)(25,28,31,39)(26,33,29,38)(27,30,37,36)(32,40,35,34);(1,3,4,8)(2,6,5,7)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,34)(35,40);(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(25,29,31,26)(27,32,37,35)(28,38,39,33)(30,40,36,34);(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,30,31,36)(26,34,29,40)(27,28,37,39)(32,38,35,33)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575557);