Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#3283 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C3×#3283 | C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C4×D4×C9 ◁ G | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,20,17)(16,24,27,29)(18,23,19,22)(21,28,25,26);(14,16)(15,21)(17,25)(18,26)(19,28)(20,27)(22,24)(23,29);(11,13)(14,17,20,15)(16,25,27,21)(18,23,19,22)(24,26,29,28);(14,18)(15,22)(17,23)(19,20)(24,29)(26,28)315 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2015);