Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C26 |
| direct product | C9×#1440 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C9 | C23 |
| commutator | C23 | C22×C4×C9 |
| Frattini, socle, cosocle | C24×C3 | C23×C3 |
| prime radical | C25×C3 | C22×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C9 ◁ G | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(15,17)(18,19,20,21)(22,23,25,24);(11,13)(15,17)(23,24);(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,24,25,23)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 172);