Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C4×#3507 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4);(7,11)(9,12)(13,14)(15,22)(16,17)(18,19)(20,25)(21,26)(23,28)(24,27);(5,6,8,10)(7,9,11,12)(14,19)(16,21)(20,22)(23,27);(5,7,8,11)(6,9,10,12)(13,15,18,25)(14,20,19,22)(16,23,21,27)(17,24,26,28);(13,16,18,21)(14,17,19,26)(15,23,25,27)(20,24,22,28);(13,17)(14,21)(15,24)(16,19)(18,26)(20,27)(22,23)(25,28)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532848);