Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62353 |
| 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 | |
6862 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)(13,19)(15,23)(17,24)(18,22)(20,25)(21,26);(3,5)(4,7)(6,10)(8,9)(11,12,16,14)(13,17,21,20)(15,18,23,22)(19,24,26,25);(11,13,15,19,16,21,23,26)(12,17,18,24,14,20,22,25);(5,10)(7,9)(11,14,16,12)(13,20,21,17)(15,22,23,18)(19,25,26,24);(4,8)(7,9)(12,14)(13,21)(18,22)(19,26)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484843);