Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | C22×#2327 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(6,8)(9,10,12,11)(14,18)(15,21)(16,24)(26,28);(9,11,12,10)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27);(5,6,7,8)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28);(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532496);