Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
| direct product | C2×#1299 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | #95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4,6,9,7,10,5,8)(12,15)(13,17);(4,8,10,9)(11,12,13,16,14,15,17,18)76 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1299, #105445, #105446, … |
| maximal normal subgroup of | C4×#1299, #105445, #105446, … |
| maximal quotient of | C2×#7042, #95259, #95261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1047);