Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C8 |
| direct product | C2×#1450 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C22 |
| commutator | C2 | C22×C4×C8 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C2×C42 |
| prime radical | C24 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
491 conjugacy classes of subgroups; only the 407 normal subgroups are shown.
This group has 491 conjugacy classes of subgroups — too many to draw. Only its 407 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);(11,12,14,17,15,18,21,24)(13,16,20,23,19,22,25,26);(3,4,5,7,6,8,9,10)(11,13)(12,16)(14,20)(15,19)(17,23)(18,22)(21,25)(24,26);(13,19)(16,22)(20,25)(23,26)160 conjugacy classes — view the full character table.
| maximal subgroup of | #105257, #105259, #105277, … |
| maximal normal subgroup of | #105257, #105259, #105277, … |
| maximal quotient of | C2×#7841, C4×#1450, C2×#7849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10301);