#9434196

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

4198 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:

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

Polycyclic presentation:

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

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, 9341392);