Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C29×C3 |
| direct product | C22×C32×#137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C25×C32 |
| prime radical | C22×C32×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C23×C32×C4 ◁ G | |
4392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2)(21,22,23);(3,4)(24,25,26);(5,6)(7,13)(8,16)(9,11)(10,14)(12,18)(15,20)(17,19);(5,7,9,15)(6,10,11,19)(8,17,18,14)(12,20,16,13);(6,11)(10,19)(12,16)(13,20)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4639);