Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#2255 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
259 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 259 conjugacy classes of subgroups — too many to draw. Only its 143 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);(3,4,6,8)(5,9,10,7)(12,16)(13,18)(14,20)(17,22)(19,26)(21,25)(23,24);(3,5,6,10)(4,7,8,9)(12,17)(16,22)(19,23)(24,26);(11,12,18,24,14,19,21,22,15,17,25,26,20,23,13,16)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137525, #137526, #141553, … |
| maximal normal subgroup of | #137525, #137526, #141553, … |
| maximal quotient of | C2×#12908, C2×#13167, C2×#13168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26715);