Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C8 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#2117 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C8 | C2×D4 |
| commutator | C4 | C22×C32×C8 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle | C2×C32 | C2×C22⋊C8 |
| prime radical | C2×C32×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C8 ◁ C22×C32×C8 ◁ G | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2,3)(7,8,11,17,12,18,26,35,13,15,27,31,10,16,24,33)(9,19,21,36,20,29,37,34,22,32,25,38,23,14,28,30);(4,5,6)(7,9)(8,14)(10,23)(11,25)(12,20)(13,22)(15,29)(16,32)(17,34)(18,19)(21,27)(24,37)(26,28)(30,31)(33,36)(35,38);(8,15)(14,29)(16,18)(17,31)(19,32)(30,34)(33,35)(36,38)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3177);