#9210342

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

4390 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)(3,5)(4,8)(6,7)(9,10)(11,15)(12,19)(13,16)(14,17)(18,23)(20,24)(21,22)(25,26,31,30)(27,34,29,32)(28,33,36,37)(35,40,38,39);(2,7)(3,8)(9,11,14,12)(10,15,17,19)(13,18,22,20)(16,23,21,24)(25,27,31,29)(26,32,30,34)(28,35,36,38)(33,39,37,40);(1,3,6,8)(2,4,7,5)(11,12)(15,19)(18,20)(23,24)(28,36)(33,37)(35,38)(39,40);(1,4,6,5)(2,3,7,8)(9,12,14,11)(10,15,17,19)(13,20,22,18)(16,23,21,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,36)(32,39)(34,40);(1,2)(3,4)(5,8)(6,7)(9,10)(11,15)(12,19)(13,21)(14,17)(16,22)(18,24)(20,23)(25,29,31,27)(26,32,30,34)(28,35,36,38)(33,40,37,39);(1,5,6,4)(2,8,7,3)(9,13,14,22)(10,16,17,21)(11,18,12,20)(15,23,19,24)(25,30,31,26)(27,34,29,32)(28,37,36,33)(35,40,38,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,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, 9117538);