Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#11436 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C2×M16 | C24 |
| socle | C22 | C2×#323 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,11,7,12,18,26,8,10,17,23,19,24,30,33)(5,13,15,25,16,27,29,34,14,9,20,21,28,22,31,32);(3,5)(4,9)(6,15)(7,16)(8,14)(10,13)(11,25)(12,22)(17,20)(18,29)(19,28)(21,23)(24,27)(26,34)(30,31)(32,33);(5,14)(9,13)(15,20)(16,28)(21,25)(22,27)(29,31)(32,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387201);