Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#23104 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,6,5)(8,12)(13,14)(15,16,18,22)(17,24,25,20)(19,21,27,23)(26,28,30,29);(3,5,6,4)(7,8,11,12)(9,13,10,14)(15,17)(16,20)(18,25)(19,26)(21,29)(22,24)(23,28)(27,30);(7,9,11,10)(8,13,12,14)(16,21)(17,25)(19,27)(20,28)(22,23)(24,29);(7,10,11,9)(8,13,12,14)(17,25)(20,24)(26,30)(28,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289374);