Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#22245 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(4,6)(7,8)(9,16)(10,14)(11,20)(12,15)(13,18)(17,22)(19,21)(23,24,29,25,28,31,27,33)(26,30,35,32,36,38,37,34);(7,9)(8,13)(10,19)(11,17)(12,18)(14,22)(15,16)(20,21)(23,25,29,31,28,33,27,24)(26,34,35,30,36,32,37,38);(3,4,5,6)(7,10,12,11)(8,14,15,20)(9,17,18,19)(13,21,16,22)(23,26,28,36)(24,30,31,38)(25,32,33,34)(27,37,29,35);(7,11,12,10)(8,14,15,20)(9,17,18,19)(13,22,16,21)(23,27,28,29)(24,25,31,33)(26,35,36,37)(30,34,38,32)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283530);