Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C29×C3 |
| direct product | C23×C32×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C32 | D4 |
| commutator, Frattini, cosocle | C4 | C25×C32 |
| prime radical | C23×C32×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C32 ◁ C23×C32×C4 ◁ G | |
5160 conjugacy classes of subgroups; only the 2856 normal subgroups are shown.
This group has 5160 conjugacy classes of subgroups — too many to draw. Only its 2856 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(15,16,17);(3,4)(18,19,20);(5,6);(7,8,9,10,11,12,13,14);(8,10)(9,13)(12,14)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4635);