Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C25 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1439 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1439 conjugacy classes of subgroups — too many to draw. Only its 567 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2);(3,4);(5,6)(7,8)(9,10,13,16)(11,17,19,24)(12,21,18,15)(14,22,23,20);(9,11,18,22,13,19,12,20)(10,14,21,24,16,23,15,17);(6,8)(11,19)(14,23)(17,24)(20,22)64 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#470 |
| maximal subgroup of | #111401, #111404, #129714, … |
| maximal normal subgroup of | #111401, #111404, #129714, … |
| maximal quotient of | C22×#1689, C22×#1695, C22×#1703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53428);