Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#504 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C16 | C22 |
| commutator | C2 | C22×C8×C9 |
| Frattini, cosocle | C3×C8 | C23×C3 |
| socle | C2×C3 | C22×C3×C8 |
| prime radical | C3×#41 | C3×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C9×C16 ◁ G | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,17,14,18,23,29,15,19,24,30,25,31,35,39)(12,16,21,27,22,28,34,38,20,26,32,36,33,37,40,41);(10,12)(11,16)(13,21)(14,22)(15,20)(17,27)(18,28)(19,26)(23,34)(24,32)(25,33)(29,38)(30,36)(31,37)(35,40)(39,41);(12,20)(16,26)(21,32)(22,33)(27,36)(28,37)(34,40)(38,41)360 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 940);