Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1891 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | #120 |
| commutator | C23 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#386 |
| prime radical | C3×#525 | C2×C3 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C22×C4×C9 ◁ G | |
513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,14,18,15,12,16,20)(13,19,22,25,21,17,23,24);(10,12,14,20,15,11,16,18)(13,19,23,24,21,17,22,25);(10,13)(11,17)(12,19)(14,23)(15,21)(16,22)(18,25)(20,24)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 623);