Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#1864 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,6)(7,8,10,9)(11,12,15,18)(13,16,14,17);(4,6)(7,9,10,8)(12,16)(13,14)(17,18);(12,17)(13,14)(16,18)76 conjugacy classes — view the full character table.
| maximal subgroup of | #99887, #99903, C4×#1864, … |
| maximal normal subgroup of | #99887, #99903, C4×#1864, … |
| maximal quotient of | C2×#8845, C2×#8849, #191889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13857);