Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1986 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(12,15,16,17);(4,7)(8,9)(11,12,13,17,14,16,18,15);(3,5,6,10)(4,8,7,9)76 conjugacy classes — view the full character table.
| maximal subgroup of | #107437, #107439, #107529, … |
| maximal normal subgroup of | #107437, #107439, #107529, … |
| maximal quotient of | C2×#9864, C2×#9865, C2×#9866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14293);