Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#23091 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C2×#392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2602 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)(5,6)(8,12)(13,14)(15,16,20,22,19,23,18,27)(17,21,25,30,26,24,28,29);(7,8,11,12)(9,13,10,14)(15,16,18,27,19,23,20,22)(17,24,28,30,26,21,25,29);(4,6)(7,9,11,10)(8,13,12,14)(15,17,19,26)(16,21,23,24)(18,28,20,25)(22,30,27,29);(7,10,11,9)(8,13,12,14)(15,18,19,20)(16,22,23,27)(17,25,26,28)(21,29,24,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289336);