Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1492 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
715 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)(5,7)(6,9)(8,10)(11,12,14,17,15,13,18,16);(4,7)(5,10)(8,9)(11,13)(12,15)(14,16)(17,18);(12,13)(16,17)88 conjugacy classes — view the full character table.
| maximal subgroup of | #108084, #108088, #108341, … |
| maximal normal subgroup of | #108084, #108088, #108341, … |
| maximal quotient of | C2×#7658, C2×#7659, C2×#7660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10458);