Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#3414 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C25 |
| socle | C22 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1049 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 1049 conjugacy classes of subgroups — too many to draw. Only its 479 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) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,14)(6,12)(7,19)(8,23)(9,11)(10,16)(13,21)(15,31)(17,30)(18,28)(20,26)(22,33)(24,29)(25,32)(27,34);(3,5)(4,10)(6,15)(7,20)(8,24)(9,17)(11,25)(12,26)(13,29)(14,19)(16,21)(18,32)(22,34)(23,27)(28,31)(30,33);(4,11)(10,25)(12,23)(13,28)(14,30)(19,33)(26,27)(29,31);(3,6,9,8)(4,12,11,23)(5,15,17,24)(7,18,22,21)(10,26,25,27)(13,19,28,33)(14,31,30,29)(16,20,32,34)64 conjugacy classes — view the full character table.
| maximal subgroup of | #147904, #147905, #147906, … |
| maximal normal subgroup of | #147904, #147905, #147906, … |
| maximal quotient of | C2×#33380, C2×#33409, C2×#33451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55619);