Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#60343 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C22×2−1+4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4,8,9,7,6,5,10)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26);(3,5,7,8)(4,9,6,10)(11,13,15,21)(12,16,18,19)(14,22,23,20)(17,25,26,24);(3,5,7,8)(4,10,6,9)(11,14,15,23)(12,17,18,26)(13,20,21,22)(16,24,19,25);(12,18)(16,19)(17,26)(24,25);(3,6,8,10,7,4,5,9)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481748);