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) = C25 |
| direct product | C2×#1485 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
537 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 537 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,7,10,8,5,6,9)(11,12)(13,20)(14,18)(15,24)(16,19)(17,22)(21,25)(23,26);(3,5,8,4)(6,10,7,9)(11,13,16,23)(12,17,19,25)(14,22,15,21)(18,26,24,20);(3,6,8,7)(4,9,5,10)(11,14,16,15)(12,18,19,24)(13,21,23,22)(17,20,25,26)76 conjugacy classes — view the full character table.
| maximal subgroup of | #107443, #107446, #107883, … |
| maximal normal subgroup of | #107443, #107446, #107883, … |
| maximal quotient of | C2×#7675, C2×#7680, C2×#7682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10448);