Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C27 |
| direct product | C9×#2308 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C22 | C23×C4×C9 |
| Frattini, socle, cosocle | C23×C3 | C24×C3 |
| prime radical | C23×C3×D4 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
1740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(11,13)(14,15,16,17)(18,19,21,22)(20,24,25,23);(19,22)(23,24);(10,11)(12,13)(14,15)(16,17)(18,19)(20,23)(21,22)(24,25);(18,20)(19,23)(21,25)(22,24)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1040);