Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#401 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C23 |
| commutator | C23 | C23×C9 |
| Frattini, socle, prime radical, cosocle | C23×C3 | C23×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(12,16)(14,17)(19,22)(23,24)(26,27,29,31)(28,32,33,30);(10,11,13,15)(12,17,16,14)(18,19,21,22)(20,24,25,23)(26,27)(28,32)(29,31)(30,33);(10,12)(11,14)(13,16)(15,17)(18,20,21,25)(19,23,22,24)(26,28,29,33)(27,30,31,32)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 837);