Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C2×C9×#430 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C2×D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C2×C3×C4⋊C4 |
| prime radical | C24×C3 | C3×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C23×C4×C9 ◁ G | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(3,4,7,10,8,11,6,16)(5,12,15,17,13,18,14,9);(3,5,6,14,8,13,7,15)(4,9,10,12,11,17,16,18);(5,13)(9,17)(12,18)(14,15)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1647);