#9523439

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

4371 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9430635);