Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#2163 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | C22×C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 223 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)(5,8)(6,7)(9,10)(11,12,13,17,14,18,19,24,15,16,20,22,21,23,25,26);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(12,16)(17,22)(18,23)(24,26)112 conjugacy classes — view the full character table.
| maximal subgroup of | #126562, #126563, #126571, … |
| maximal normal subgroup of | #126562, #126563, #126571, … |
| maximal quotient of | C2×#12591, C2×#12599, C2×#12653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26404);