Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C27 · Schur multiplier (H2) = C25 |
| direct product | C2×C27×#95 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C27 | C22 |
| commutator | C2 | C2×C42×C27 |
| Frattini, cosocle | C23×C9 | C23×C3 |
| socle, prime radical | C24×C3 | C22×C9 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C27 ◁ G | |
660 conjugacy classes of subgroups; only the 420 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 420 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(8,10)(11,12,14,13);(3,4,5,6)(7,8,9,10)(11,13,14,12)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 817);