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 | C2×C32×#575 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C25×C32 |
| socle | C22×C32 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C23×C32×C4 ◁ G | |
4452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(19,20,21);(3,4)(5,12)(6,11)(7,15)(8,10)(9,13)(14,18)(16,17)(22,23,24);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15);(4,10)(9,17)(11,15)(12,18);(3,6,8,7)(4,11,10,15)(5,13,14,16)(9,18,17,12)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4640);