Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#1384 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #106 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | #340 |
| prime radical | C24 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
293 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 159 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 — 3 generators on 34 points:
(1,2);(3,4)(5,15,17,29)(6,12)(7,22)(8,24)(9,14)(10,16,25,30)(11,20)(13,23)(18,33,31,27)(19,34,32,26)(21,28);(3,5,7,18,23,32,21,30,9,17,20,31,8,19,6,16)(4,10,12,26,24,33,11,15,14,25,28,34,13,27,22,29)64 conjugacy classes — view the full character table.
| maximal subgroup of | #94347, #94349, C4×#1384, … |
| maximal normal subgroup of | #94347, #94349, C4×#1384, … |
| maximal quotient of | C2×#7157, #126656, #126693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5552);