Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,21,23)(25,26,32,37)(27,30,38,33)(28,35,29,34)(31,36,39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,20)(18,22)(23,24)(25,27,32,38)(26,33,37,30)(28,31,29,39)(34,36,35,40);(2,3)(6,7)(13,22)(16,24)(18,21)(20,23)(25,28,32,29)(26,34,37,35)(27,31,38,39)(30,40,33,36);(4,8)(6,7)(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,24,21)(25,29,32,28)(26,35,37,34)(27,31,38,39)(30,40,33,36);(1,3,5,2)(4,7,8,6)(9,13,14,22)(10,16,17,24)(11,18,19,21)(12,20,15,23)(25,30,32,33)(26,27,37,38)(28,36,29,40)(31,34,39,35);(1,4)(2,6)(3,7)(5,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,22)(23,24)(25,31)(26,36)(27,29)(28,38)(30,34)(32,39)(33,35)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8519393);