Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1503 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
597 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,7)(5,9)(8,10)(12,16,17,18);(3,4,6,8)(5,10,9,7)(11,12)(13,16)(14,18)(15,17);(11,13,15,14)(12,16,17,18)76 conjugacy classes — view the full character table.
| maximal subgroup of | #106036, #107703, #107713, … |
| maximal normal subgroup of | #106036, #107703, #107713, … |
| maximal quotient of | C2×#7562, C2×#7570, C2×#7571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10520);