Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#2231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
263 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 263 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);(3,4,9,14)(5,15,18,13)(6,19)(7,23,16,12)(8,10,25,27)(11,28)(17,31)(20,33)(21,32)(22,26)(24,29)(30,34);(3,5,16,25,9,18,7,8)(4,10,23,13,14,27,12,15)(6,17,21,34,22,28,20,24)(11,26,30,32,31,19,29,33);(3,6)(4,11)(5,17)(7,20)(8,24)(9,22)(10,26)(12,29)(13,32)(14,31)(15,33)(16,21)(18,28)(19,27)(23,30)(25,34)46 conjugacy classes — view the full character table.
| maximal subgroup of | #138569, #138571, #138573, … |
| maximal normal subgroup of | #138569, #138571, #138573, … |
| maximal quotient of | C2×#12824, C2×#12950, C2×#13116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26624);