Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#1420 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #357 |
| commutator | C16 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C16 | C22×C3 |
| socle | C2×C3 | C3×#357 |
| prime radical | C22×C3 | C3×SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C2×C8×C9 ◁ C2×C9×C16 ◁ G | |
105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 73 points:
(1,2,3,4,5,6,7,8,9)(10,11,14,20,17,22,16,40)(12,23,25,54,28,56,27,63)(13,29,32,64,35,66,34,71)(15,36,31,52,39,47,38,21)(18,41,42,24,44,58,50,60)(19,45,48,70,49,69,65,33)(26,61,57,46,62,73,37,55)(30,51,67,72,59,53,68,43);(10,12,17,28)(11,18,22,44)(13,30,35,59)(14,27,16,25)(15,37,39,57)(19,46,49,55)(20,50,40,42)(21,51,52,53)(23,29,56,66)(24,33,60,70)(26,31,62,38)(32,68,34,67)(36,43,47,72)(41,45,58,69)(48,61,65,73)(54,71,63,64)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 152);