Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#2153 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C9 | C2×D4 |
| commutator | C4 | C22×C8×C9 |
| Frattini, cosocle | C2×C3×C8 | C23×C3 |
| socle | C2×C3 | C2×C3×C4⋊C8 |
| prime radical | C23×C3 | C3×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C9 ◁ C22×C8×C9 ◁ G | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,14,19,15,18,28,33,16,20,29,34,13,26,27,39)(12,21,24,36,25,31,38,40,22,35,37,41,23,17,30,32);(10,12,13,23,16,22,15,25)(11,17,18,21,20,31,26,35)(14,24,27,30,29,37,28,38)(19,32,33,36,34,40,39,41);(12,22)(17,31)(21,35)(23,25)(24,37)(30,38)(32,40)(36,41)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 885);