Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#1349 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 71 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 — 3 generators on 26 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,14,19,16,21,15,20)(13,22,24,18,25,17,23,26);(3,5)(4,7)(6,10)(8,9)(12,17)(13,23)(14,15)(18,19)(20,26)(21,22)(24,25)58 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1349, #128780, #128869, … |
| maximal normal subgroup of | C4×#1349, #128780, #128869, … |
| maximal quotient of | C2×#7080, C2×#7083, #122582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4695);