Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#1389 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #102 |
| commutator | C8 | C32×C42 |
| Frattini, cosocle | C22×C8 | C22×C32 |
| socle | C2×C32 | #340 |
| prime radical | C23×C32 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 38 points:
(1,2,3)(7,8,11,16,13,17,12,28)(9,18,20,35,23,31,22,37)(10,24,25,33,27,38,26,15)(14,29,30,21,32,36,34,19);(4,5,6)(7,9,10,21,11,20,25,36,13,23,27,19,12,22,26,29)(8,14,15,31,28,34,38,35,17,32,33,18,16,30,24,37)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2449);