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×#59270 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6244 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)(12,15)(13,17)(19,20,23,25)(21,22,26,24);(3,4,8,9)(5,10,6,7)(12,15)(16,18)(21,26)(22,24);(3,5,8,6)(4,10,9,7)(11,12)(13,16)(14,15)(17,18)(19,21)(20,24)(22,25)(23,26);(3,6,8,5)(4,7,9,10)(11,13,14,17)(12,16,15,18)(19,22,23,24)(20,21,25,26);(3,7)(4,5)(6,9)(8,10)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573501);