Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C42 |
| direct product | C2×#14155 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(4,6)(7,8,9,10)(16,18)(17,21);(11,12,13,14)(15,16,19,18)(17,20,21,22);(3,4,5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,20)(18,22)(19,21);(15,18,19,16)(17,22,21,20)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421799);