Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10362 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#10368 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #393 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,5,7)(6,8,10,9)(11,12,14,17)(13,20)(15,19,16,24)(18,21)(22,25)(23,26);(4,7)(8,9)(12,17)(13,21,23,22)(18,26,25,20)(19,24);(4,8)(6,10)(7,9)(11,13)(12,18)(14,23)(15,21)(16,22)(17,25)(19,26)(20,24)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145135);