Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#3363 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
915 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 915 conjugacy classes of subgroups — too many to draw. Only its 503 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 — 5 generators on 26 points:
(1,2);(4,7)(8,9)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20);(3,4,6,7)(5,9,10,8);(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19);(3,5,6,10)(4,8,7,9)70 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#3363⋊C3 |
| maximal subgroup of | #220381, #220453, #220753, … |
| maximal normal subgroup of | #220381, #220453, #220753, … |
| maximal quotient of | C2×Q8×#103, C2×#22542, C2×#22740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54288);