#9210351

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

4366 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)(27,35)(28,37)(29,34)(30,33)(31,32)(36,39)(38,40);(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)(26,29)(28,36)(30,40)(31,35)(32,34)(33,39)(37,38);(1,3,6,5)(2,8,4,7)(11,12)(15,19)(18,20)(23,24)(25,28,32,30)(26,33,31,37)(27,36,34,40)(29,39,35,38);(9,12,14,11)(10,15,17,19)(13,20,22,18)(16,23,21,24)(25,29,32,35)(26,34,31,27)(28,38,30,39)(33,36,37,40);(1,4,6,2)(3,7,5,8)(9,10)(11,15)(12,19)(13,21)(14,17)(16,22)(18,24)(20,23)(25,30,32,28)(26,33,31,37)(27,36,34,40)(29,38,35,39);(1,5,6,3)(2,7,4,8)(9,13,14,22)(10,16,17,21)(11,18,12,20)(15,23,19,24)(25,31,32,26)(27,35,34,29)(28,37,30,33)(36,38,40,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gi,e^2=h,f^2=hi,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, 9117547);