Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61684 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,17)(14,19)(15,18)(16,21)(20,24)(22,25)(23,26);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,16)(15,22)(18,25)(19,21)(20,26)(23,24);(12,18)(15,24)(17,25)(19,26)(20,21)(22,23);(5,10)(7,9)(11,14,16,13)(12,19,21,17)(15,23,24,22)(18,26,20,25);(4,8)(7,9)(13,14)(17,19)(22,23)(25,26)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484486);