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×#444 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C2×D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C3×C4×D4 |
| prime radical | C2×C3×#135 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C22×C8×C9 ◁ G | |
696 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);(3,4,7,11,8,5,12,10)(6,13,14,17,15,9,16,18);(3,5)(4,8)(6,13)(7,10)(9,15)(11,12)(14,17)(16,18);(3,6,7,14,8,15,12,16)(4,9,11,18,5,13,10,17)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1686);