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×C4 |
| direct product | C2×#21824 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8)(9,15)(10,13)(11,14)(12,17)(16,21)(18,20)(19,22)(23,24,26,29,27,30,33,36)(25,28,32,35,31,34,37,38);(7,9,11,17)(8,12,14,15)(10,18,19,16)(13,21,22,20)(23,25)(24,28)(26,32)(27,31)(29,35)(30,34)(33,37)(36,38);(4,6)(7,10,11,19)(8,13,14,22)(9,16,17,18)(12,20,15,21);(8,14)(12,15)(13,22)(20,21)(25,31)(28,34)(32,37)(35,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281521);