Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C2×C9×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | D4 |
| commutator | C4 | C24×C9 |
| Frattini, cosocle | C3×C4 | C24×C3 |
| socle | C22×C3 | C2×C3×D4 |
| prime radical | C2×C3×#135 | C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C22×C4×C9 ◁ G | |
399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(3,4)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1886);