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×#59968 |
| 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 | |
5714 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);(4,7)(8,10)(12,16)(13,18)(19,20)(21,24)(22,26)(23,25);(3,4)(5,8)(6,7)(9,10)(12,16)(13,18)(20,21)(25,26);(11,12)(13,17)(14,18)(15,16)(19,21,24,20)(22,26,23,25);(3,5,6,9)(4,8,7,10)(11,13,15,18)(12,17,16,14)(19,22,24,23)(20,25,21,26);(3,4)(5,8)(6,7)(9,10)(11,14,15,17)(12,18,16,13)(19,23,24,22)(20,26,21,25)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573785);