Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#1591 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle, prime radical | C23 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
335 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 183 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 — 4 generators on 26 points:
(1,2);(3,4,7,5,6,8,10,9)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20);(4,8)(5,9)(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19);(3,5,7,8,6,9,10,4)88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7073, #122491, #122499, … |
| maximal normal subgroup of | C2×#7073, #122491, #122499, … |
| maximal quotient of | C2×#8207, C2×#8282, C4×#1591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10873);