Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#441 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C2×D4 |
| commutator | C4 | C22×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C3 |
| socle, prime radical | C22×C3 | C2×C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C42×C9 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,12)(14,15)(16,22)(17,25)(18,20)(19,23)(21,27)(24,29)(26,28);(14,16,18,24)(15,19,20,28)(17,26,27,23)(21,29,25,22);(10,12,13,11)(15,20)(19,28)(21,25)(22,29)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 877);