Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C32 |
| prime radical | C2×C32×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ G | |
336 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(8,10)(11,12)(13,19)(14,16)(15,23)(17,21)(18,22)(20,26)(24,25);(4,5,6)(7,8,9,10)(12,16)(17,25)(18,23)(19,26);(11,13,14,20)(12,17,16,25)(15,24,22,21)(18,26,23,19)171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4919);