Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C2×C9×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C9 | C22 |
| commutator | C2 | C23×C4×C9 |
| Frattini, cosocle | C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×C4 |
| prime radical | C2×C3×#41 | C2×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×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);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18);(5,11)(8,14)(12,17)(15,18)360 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1881);