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×#62351 |
| 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 | |
6748 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)(13,19)(14,17)(15,23)(16,22)(18,20)(21,26)(24,25);(3,5)(4,7)(6,10)(8,9)(11,12,14,17)(13,16,21,24)(15,18,20,23)(19,25,26,22);(11,13,20,25,14,21,15,22)(12,16,23,26,17,24,18,19);(5,10)(7,9)(11,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,26);(4,8)(7,9)(12,17)(13,21)(18,23)(22,25)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484841);