Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#2860 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1419 conjugacy classes of subgroups; only the 559 normal subgroups are shown.
This group has 1419 conjugacy classes of subgroups — too many to draw. Only its 559 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(4,6)(8,9)(12,13)(17,20)(18,22);(7,8,11,9)(10,12,14,13)(15,16,19,21)(17,18,20,22);(3,4,5,6)(7,9,11,8)(10,13,14,12)(15,17,19,20)(16,18,21,22);(7,10,11,14)(8,12,9,13)(15,18)(16,20)(17,21)(19,22)88 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×C4⋊(#126⋊C3) |
| maximal subgroup of | #96933, C2×#7666, C2×#7672, … |
| maximal normal subgroup of | #96933, C2×#7666, C2×#7672, … |
| maximal quotient of | C2×#14688, C2×#14699, C2×#14700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29617);