Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2223 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(4,10)(5,16)(6,19)(7,8)(11,20)(12,27)(13,24)(14,23)(15,25)(17,32)(18,21)(22,28)(29,34)(30,33);(3,4,11,23,8,14,6,10,9,15,19,24,7,13,20,25)(5,12,26,34,16,28,17,22,18,29,31,27,21,30,32,33);(3,5)(4,12)(6,17)(7,21)(8,16)(9,18)(10,22)(11,26)(13,30)(14,28)(15,29)(19,31)(20,32)(23,34)(24,27)(25,33)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2223, #495890, #498371, … |
| maximal normal subgroup of | C4×#2223, #495890, #498371, … |
| maximal quotient of | C2×#12432, C2×#12435, #490476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26604);