Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×#1798 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C25 | C22×C4 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2);(3,4);(6,8)(9,10)(11,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(5,6,7,8)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(10,12)(15,21)(18,24)(22,27)(25,28)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277956);