Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#18043 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,6,9,7,5,8,10)(11,12)(13,20)(14,19)(15,23)(16,18)(17,21)(22,26)(24,25);(11,13,16,22)(12,17,18,25)(14,21,15,24)(19,26,23,20);(3,5,6,10,7,4,8,9)(12,18)(17,25)(19,23)(20,26);(4,5)(9,10)(11,14,16,15)(12,19,18,23)(13,21,22,24)(17,26,25,20)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252224);