#9523521

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

4299 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)(11,16)(12,18)(13,19)(14,21)(15,17)(20,24)(22,23)(25,26,31,29)(27,28,35,32)(30,37,38,33)(34,39,40,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,21)(13,15)(14,18)(17,19)(20,22)(23,24)(25,27,31,35)(26,32,29,28)(30,34,38,40)(33,39,37,36);(1,2,4,6)(3,7,8,5)(10,17)(11,13)(12,22)(14,24)(26,29)(28,32)(30,38)(34,40);(3,8)(5,7)(9,12,15,22)(10,18,17,23)(11,21,13,20)(14,19,24,16)(25,28)(26,27)(29,35)(30,36)(31,32)(33,34)(37,40)(38,39);(1,3,4,8)(2,5,6,7)(9,13,15,11)(10,19,17,16)(12,21,22,20)(14,23,24,18)(25,29,31,26)(27,32,35,28)(30,37,38,33)(34,39,40,36);(1,2)(3,5)(4,6)(7,8)(9,14)(10,20)(11,18)(12,19)(13,23)(15,24)(16,22)(17,21)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=gh,e^2=gi,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, 9430717);