Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1818 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C24 | C22×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
763 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 763 conjugacy classes of subgroups — too many to draw. Only its 271 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) = 128 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16,18,21,19,22,25,28)(17,20,24,27,23,26,29,30);(8,11)(12,14)(15,17)(16,20)(18,24)(19,23)(21,27)(22,26)(25,29)(28,30);(4,6)(7,9,10,13)(8,12,11,14)(17,23)(20,26)(24,29)(27,30)88 conjugacy classes — view the full character table.
| maximal subgroup of | #107148, #107153, #108234, … |
| maximal normal subgroup of | #107148, #107153, #108234, … |
| maximal quotient of | C2×#8702, C2×#8703, #189697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13669);