Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C26 |
| direct product | C9×#3009 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C2×C3×H(5,𝔽2) | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C9×#41 ◁ G | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,18,22)(16,20,23,27)(17,19,25,21)(24,26,29,28);(15,19)(17,25)(20,26)(21,22)(24,29)(27,28);(11,13)(14,16)(15,20)(17,24)(18,23)(19,26)(21,28)(22,27)(25,29);(16,23)(20,27)(24,29)(26,28)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1741);