Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#2726 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2);(3,4);(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,23,20);(6,8)(10,11)(12,16)(17,19,21,23)(18,22,24,20);(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,20,21,22)(18,23,24,19);(9,12,13,16)(10,14,11,15)(17,18)(19,22)(20,23)(21,24)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540685);