Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#1353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 64 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(4,7,8,9)(11,12,14,18,15,17,21,24)(13,16,20,22,19,23,26,25);(3,4)(5,9)(6,8)(7,10)(11,13)(12,16,17,23)(14,20)(15,19)(18,25,24,22)(21,26)46 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1353, #128790, #128817, … |
| maximal normal subgroup of | C4×#1353, #128790, #128817, … |
| maximal quotient of | C2×#7122, C2×#7132, #124192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4707);