#9164000

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

4391 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9071196);