Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | C42 |
| commutator | C4 | C42×C9 |
| Frattini, cosocle | C22×C3×C4 | C22×C3 |
| socle | C2×C3 | C3×#95 |
| prime radical | C23×C3 | C3×D4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ G | |
141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,18,20,24)(13,16)(14,17)(15,19,22,25)(21,23);(10,12)(11,15,17,22)(13,19)(14,20)(16,18,23,24)(21,25)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 143);