Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#59731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5698 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);(4,9)(7,10)(11,12,16,15)(13,14,18,17)(20,22)(21,25);(3,4,8,9)(5,10,6,7)(13,18)(14,17)(19,20,23,22)(21,26,25,24);(3,5,8,6)(4,10,9,7)(11,13,16,18)(12,17,15,14)(19,21,23,25)(20,24,22,26);(3,6,8,5)(4,7,9,10)(11,14)(12,13)(15,18)(16,17)(19,20,23,22)(21,24,25,26);(3,7)(4,5)(6,9)(8,10)(11,15,16,12)(13,14,18,17)(19,22,23,20)(21,26,25,24)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573800);