Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#42740 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2414 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5446 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);(3,4,7,8)(5,6,10,9)(11,12,15,17)(13,14,18,16)(19,20,23,26)(21,24,22,25);(4,8)(6,9)(11,13,15,18)(12,16,17,14)(19,21,23,22)(20,24,26,25);(4,8)(5,10)(11,13)(12,14)(15,18)(16,17)(19,20)(21,24)(22,25)(23,26);(3,5,7,10)(4,9,8,6)(11,14)(12,13)(15,16)(17,18)(19,22,23,21)(20,25,26,24);(3,6,7,9)(4,10,8,5)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548413);