Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1890 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #120 |
| commutator | C23 | C22×C4×C9 |
| Frattini, cosocle | C24×C3 | C23×C3 |
| socle | C22×C3 | C3×#120 |
| prime radical | C22×C3×D4 | C22×C3 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C24×C9 ◁ G | |
579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(11,13)(14,15)(16,19)(17,24,25,20)(18,21)(22,26,28,29)(23,27);(10,11,12,13)(14,16,18,23)(15,19,21,27)(17,22,25,28)(20,26,24,29);(14,17)(15,20)(16,22)(18,25)(19,26)(21,24)(23,28)(27,29)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 622);