Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×#427 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C24 | C2×D4 |
| prime radical | C25 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1503 conjugacy classes of subgroups; only the 1039 normal subgroups are shown.
This group has 1503 conjugacy classes of subgroups — too many to draw. Only its 1039 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)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(9,11,13,20,14,21,12,19)(10,15,16,18,17,23,22,24);(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)112 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#427 |
| maximal subgroup of | #123874, #123879, #123880, … |
| maximal normal subgroup of | #123874, #123879, #123880, … |
| maximal quotient of | C22×#1564, C22×#1568, C2×#17793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53091);