Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C26×C3 |
| direct product | C32×#1446 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C32 | C23 |
| commutator | C23 | C22×C32×C4 |
| Frattini, cosocle | C24 | C23×C32 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C32 ◁ G | |
2208 conjugacy classes of subgroups; only the 744 normal subgroups are shown.
This group has 2208 conjugacy classes of subgroups — too many to draw. Only its 744 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(8,10)(11,12,13,14)(16,18);(4,5,6)(15,16)(17,18)(19,20,22,24)(21,25,26,23);(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21,22,26)(20,23,24,25)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2506);