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×#10366 |
| 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)(13,20,22,26)(14,18)(15,24)(16,19)(17,23,25,21);(4,7)(8,9)(13,21,22,23)(17,20,25,26);(4,8)(6,10)(7,9)(11,13)(12,17)(14,21)(15,23)(16,22)(18,26)(19,25)(20,24)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145134);