#9515319

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

4195 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,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, 9422515);