Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11318 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #401 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(7,8)(9,12,16,20)(10,17)(11,14)(13,18)(15,19,22,21);(3,4)(5,6)(7,9)(8,12,14,20)(10,15,18,22)(11,16)(13,21)(17,19);(4,6)(7,10)(8,13,14,17)(9,15,16,22)(11,18)(12,19)(20,21)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265561);