Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31115 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,6)(7,9)(8,10)(11,12,15,19,16,20,26,17)(13,14,21,23,18,25,22,24);(3,5)(4,8)(6,9)(7,10)(11,13,16,18)(12,14,20,25)(15,21,26,22)(17,24,19,23);(3,6,7,8)(4,5,9,10)(12,17)(13,21)(15,26)(18,22)(19,20)(23,24);(4,9)(5,10)(11,14)(12,18)(13,20)(15,24)(16,25)(17,22)(19,21)(23,26)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410657);