#9683116

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9590312);