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×#3504 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C22 | C27 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
33898 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)(15,19)(16,23)(17,18)(20,24)(21,27)(22,25)(26,28);(14,18)(19,25)(20,23)(26,27);(5,7,8,12)(6,9,10,11)(13,15,17,22)(14,19,18,25)(16,21,24,28)(20,26,23,27);(13,14)(15,19)(16,20)(17,18)(21,26)(22,25)(23,24)(27,28);(13,16,17,24)(14,20,18,23)(15,21,22,28)(19,26,25,27)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493181);