Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#3531 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| socle | C23 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3277 conjugacy classes of subgroups; only the 2847 normal subgroups are shown.
This group has 3277 conjugacy classes of subgroups — too many to draw. Only its 2847 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20)(21,24)(22,29)(23,26)(25,31)(27,34)(28,33)(30,32);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(19,21,23,28)(20,24,26,33)(22,30,31,27)(25,34,29,32);(5,12)(8,11)(13,18)(15,17)(19,22,23,31)(20,25,26,29)(21,27,28,30)(24,32,33,34);(4,9)(5,12)(10,16)(13,18)(20,26)(24,33)(25,29)(32,34);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)76 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#3531⋊C5 |
| maximal subgroup of | C2×#8505, #187580, #187595, … |
| maximal normal subgroup of | C2×#8505, #187580, #187595, … |
| maximal quotient of | C2×#34350, C2×#34348, C2×C4×#562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55681);