Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#1476 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ 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);(3,4,5,7,6,8,9,10)(11,12,15,18,16,13,14,17);(11,13)(12,16)(14,18)(15,17);(11,14,16,15)(12,17,13,18)112 conjugacy classes — view the full character table.
| maximal subgroup of | #105672, #105673, #105678, … |
| maximal normal subgroup of | #105672, #105673, #105678, … |
| maximal quotient of | C2×#7560, C2×#7562, C2×#7563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10421);