Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#350 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C9 | D4 |
| commutator | C4 | C2×C8×C9 |
| Frattini, cosocle | C2×C3×C8 | C22×C3 |
| socle | C2×C3 | C3×C22⋊C8 |
| prime radical | C3×#41 | C3×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C9 ◁ C2×C8×C9 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,14,19,15,20,29,37,16,21,30,38,13,18,27,35)(12,22,24,39,23,33,40,36,25,34,28,41,26,17,31,32);(10,12)(11,17)(13,26)(14,28)(15,23)(16,25)(18,34)(19,36)(20,22)(21,33)(24,30)(27,40)(29,31)(32,38)(35,39)(37,41)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 156);