Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#38010 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5382 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);(4,5)(8,9)(12,16)(17,18)(19,20,23,25)(21,22,24,26);(3,4,7,5)(6,9,10,8)(19,21)(20,22)(23,24)(25,26);(3,4,7,5)(6,8,10,9)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,25)(23,26);(3,5,7,4)(6,9,10,8)(11,13,15,14)(12,17,16,18);(3,6)(4,8)(5,9)(7,10)(11,14,15,13)(12,18,16,17)(19,22)(20,24)(21,25)(23,26)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542434);