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×#3502 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C24 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7050 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);(5,6,8,10)(7,11,12,9)(13,14,18,22)(15,20,16,19)(17,21,25,28)(23,27,24,26);(13,15,18,16)(14,19,22,20)(17,24,25,23)(21,27,28,26);(5,7,8,12)(6,9,10,11)(13,16,18,15)(14,20,22,19)(17,24,25,23)(21,27,28,26);(13,17,18,25)(14,21,22,28)(15,23,16,24)(19,26,20,27);(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532857);