Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#485 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C32 |
| prime radical | C23×C32 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C22×C32×C4 ◁ G | |
288 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(7,8)(9,12)(10,11)(13,14)(15,16,19,23)(17,20,25,29)(18,21,26,22)(24,27,30,28);(4,5,6)(8,11)(12,14)(15,17,19,25)(16,20,23,29)(18,24,26,30)(21,27,22,28);(7,9,10,13)(8,12,11,14)(16,21)(18,26)(20,27)(22,23)(24,30)(28,29)171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4940);