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 |
| direct product | C9×#2955 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C9 | C2×D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C2×C3 | C2×C3×C4×D4 |
| prime radical | C3×#575 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C9 ◁ C22×C8×C9 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,15,20,16,19,14,28)(12,21,23,17,24,32,29,31)(13,18,26,34,27,33,25,38)(22,35,36,30,37,41,39,40);(10,12)(11,17)(13,22)(14,29)(15,23)(16,24)(18,30)(19,31)(20,32)(21,28)(25,39)(26,36)(27,37)(33,40)(34,41)(35,38);(10,13,15,26,16,27,14,25)(11,18,20,34,19,33,28,38)(12,22,23,36,24,37,29,39)(17,30,32,41,31,40,21,35);(11,19)(17,31)(18,33)(20,28)(21,32)(30,40)(34,38)(35,41)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1687);