Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#2225 |
| 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 | |
287 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 135 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)(13,19)(14,21)(17,26)(18,25)(20,23)(22,24);(3,5,6,10)(4,7,8,9);(11,12,15,18)(13,20,22,26)(14,23,21,17)(16,24,25,19)58 conjugacy classes — view the full character table.
| maximal subgroup of | #136945, #136950, #136962, … |
| maximal normal subgroup of | #136945, #136950, #136962, … |
| maximal quotient of | C2×#12812, C2×#12938, C2×#13110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26611);