Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51559 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5326 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)(5,10)(6,9)(7,8)(11,12)(13,14)(15,16)(17,18)(21,25)(23,26);(3,5,8,6)(4,9,7,10)(12,16)(14,17)(19,20,22,24)(21,23,25,26);(5,6)(9,10)(11,13)(12,17)(14,16)(15,18)(19,21)(20,23)(22,25)(24,26);(3,6,8,5)(4,9,7,10)(11,14,15,17)(12,13,16,18);(3,7,8,4)(5,9,6,10)(11,13)(12,17)(14,16)(15,18)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614805);