Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#33485 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C23×C8 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,16,20,17,13,21,23,15,18,24,25,14,22,19,26);(4,7)(8,10)(12,18)(13,22)(20,25)(23,26);(11,13,16,23,17,18,21,25,15,22,24,26,14,12,19,20);(3,5,6,9)(4,8,7,10)(11,14,15,17)(12,13,18,22)(16,19,24,21)(20,23,25,26)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529852);