#9210356

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

4444 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=i,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9117552);