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×C4 |
| direct product | C2×C9×#418 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C2×D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C23×C3 | C2×C3×D4 |
| prime radical | C23×C3×D4 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C23×C4×C9 ◁ G | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(15,16,17,18,19,20,21,22,23);(3,4,5,6)(7,8)(9,13)(10,12)(11,14);(8,12)(9,10)(13,14);(7,9,11,10)(8,13,14,12)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1625);