Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35482 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4,8,6)(5,9,10,7)(11,12,16,15)(13,18,17,14)(19,20,23,26)(21,24,22,25);(3,5)(4,9)(6,7)(8,10)(11,13)(12,14)(15,18)(16,17)(19,20)(21,24)(22,25)(23,26);(4,6)(5,10)(11,14,16,18)(12,17,15,13)(19,21,23,22)(20,24,26,25);(3,6,8,4)(5,7,10,9)(19,21,23,22)(20,24,26,25);(3,7)(4,5)(6,10)(8,9)(11,15,16,12)(13,18,17,14)(19,22,23,21)(20,25,26,24)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536048);