Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35745 |
| 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 | |
6050 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);(11,12,17,18)(13,14,16,15)(20,24)(25,26);(3,4)(5,8)(6,9)(7,10)(11,13,17,16)(12,15,18,14);(3,5)(4,7)(6,10)(8,9)(11,14,17,15)(12,13,18,16)(19,20)(21,25)(22,26)(23,24);(3,5)(4,8)(6,10)(7,9)(11,15,17,14)(12,13,18,16)(19,21,23,22)(20,25,24,26);(3,4)(5,7)(6,9)(8,10)(11,16,17,13)(12,15,18,14)(19,22,23,21)(20,26,24,25)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536489);