Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×#1363 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 256 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(4,7)(5,9)(8,10)(11,12,15,19,18,23,25,26,16,20,14,24,17,13,21,22);(3,4,6,8)(5,10,9,7)(11,13,16,23)(12,18,20,17)(14,24,15,19)(21,26,25,22)58 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1363, #128913, #128952, … |
| maximal normal subgroup of | C4×#1363, #128913, #128952, … |
| maximal quotient of | C2×#7120, C2×#7121, #124233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4921);