Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#1281 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C22 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | #95 |
| prime radical | C24 | C42 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
339 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 183 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);(3,4,5,7,6,8,9,10)(11,12)(13,19,20,16)(14,18)(15,17)(21,25,26,23)(22,24);(11,13,14,21,15,20,22,26)(12,16,18,23,17,19,24,25)88 conjugacy classes — view the full character table.
| maximal subgroup of | #93392, #93393, C4×#1281, … |
| maximal normal subgroup of | #93392, #93393, C4×#1281, … |
| maximal quotient of | C2×#7016, #94915, #94930, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 905);