Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C22×#3502 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C22 | C27 |
| prime radical | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
33442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 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, 10493158);