Finite Group DatabasebetaFinite Group Databasebeta| order | 544 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#117 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C22 |
| commutator | C2 | C22×C4×C17 |
| Frattini, cosocle | C22 | C23×C17 |
| socle | C22×C17 | C23 |
| prime radical | C23×C17 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 272 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21)(22,23)(24,27)(25,26)(28,29);(23,26)(27,29);(22,24,25,28)(23,27,26,29)340 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(544, 179);