Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3416 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C32 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C32×#41 ◁ C32×#131 ◁ G | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2,3)(7,8,12,17)(9,14,18,28)(10,13,23,27)(11,24,22,16)(15,31,26,21)(19,25,32,36)(20,35,33,30)(29,38,37,34);(4,5,6)(8,13)(10,21)(11,22)(14,25)(15,16)(17,27)(19,34)(20,33)(23,31)(24,26)(28,36)(29,30)(32,38)(35,37);(7,9)(8,14)(10,19)(11,20)(12,18)(13,25)(15,29)(16,30)(17,28)(21,34)(22,33)(23,32)(24,35)(26,37)(27,36)(31,38);(9,18)(14,28)(19,32)(20,33)(25,36)(29,37)(30,35)(34,38)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4476);