#9236501

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

4479 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143697);