Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#350 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C8 | D4 |
| commutator | C4 | C2×C32×C8 |
| Frattini, cosocle | C2×C8 | C22×C32 |
| socle | C2×C32 | C22⋊C8 |
| prime radical | C32×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C8 ◁ C2×C32×C8 ◁ G | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 38 points:
(1,2,3)(7,8,11,16,12,17,26,34,13,18,27,35,10,15,24,32)(9,19,21,36,20,30,37,33,22,31,25,38,23,14,28,29);(4,5,6)(7,9)(8,14)(10,23)(11,25)(12,20)(13,22)(15,31)(16,33)(17,19)(18,30)(21,27)(24,37)(26,28)(29,35)(32,36)(34,38)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1345);