Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#1559 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
499 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 199 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);(4,8)(5,6)(9,10)(11,12,15,20,16,19,14,18)(13,21,22,26,23,17,24,25);(3,4,7,9)(5,8,6,10)(11,13)(12,17)(14,22)(15,24)(16,23)(18,25)(19,21)(20,26);(3,5,7,6)(4,8,9,10)(11,14,16,15)(12,18,19,20)(13,22,23,24)(17,25,21,26)52 conjugacy classes — view the full character table.
| maximal subgroup of | #107351, #107609, #107625, … |
| maximal normal subgroup of | #107351, #107609, #107625, … |
| maximal quotient of | C2×#7596, C2×#7600, #160565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10721);