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×#3516 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3516 | C2 |
| 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);(6,9)(10,12)(14,17)(15,21)(20,24)(22,27);(13,14,18,17)(15,19,21,23)(16,24,25,20)(22,28,27,26);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(13,16,18,25)(14,20,17,24)(15,22,21,27)(19,26,23,28);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,23,21,19)(16,24,25,20)(22,28,27,26)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532854);