Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26196 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,8)(6,10)(7,9)(11,12)(13,22)(14,24)(15,18)(16,21)(17,20)(19,26)(23,25);(3,5,7,6)(4,8,9,10)(11,13)(12,15)(14,20)(16,24)(17,23)(18,26)(19,25)(21,22);(3,6,7,5)(4,8,9,10)(11,14)(12,18)(13,21)(15,17)(16,25)(19,23)(20,24)(22,26);(11,15)(12,13)(14,17)(16,25)(18,21)(19,24)(20,23)(22,26)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345796);