Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#3142 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1091 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1091 conjugacy classes of subgroups — too many to draw. Only its 487 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 — 5 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,22)(14,17)(15,19)(16,21)(18,25)(20,23)(24,26);(4,7)(8,10)(11,13,23,24,16,25,15,14)(12,17,19,18,21,26,20,22);(3,5,6,9)(4,8,7,10)(11,14,15,25,16,24,23,13)(12,18,20,17,21,22,19,26);(11,15,16,23)(12,19,21,20)(13,24,25,14)(17,22,26,18)64 conjugacy classes — view the full character table.
| maximal subgroup of | #112710, #112744, #112766, … |
| maximal normal subgroup of | #112710, #112744, #112766, … |
| maximal quotient of | C2×#20765, C2×#21093, C2×#21201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53605);