Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#1535 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C2×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
493 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 493 conjugacy classes of subgroups — too many to draw. Only its 183 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) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,12,8,11,6,10)(5,13,14,18,15,9,16,17)(19,20)(21,27,29,34)(22,26)(23,25)(24,28,32,30)(31,33);(3,5)(4,9)(6,14)(7,16)(8,15)(10,17)(11,13)(12,18)(19,21)(20,24)(22,30)(23,29)(25,32)(26,34)(27,33)(28,31);(3,6,8,7)(4,10,11,12)(5,14,15,16)(9,17,13,18)(20,25)(24,32)(26,33)(27,34)52 conjugacy classes — view the full character table.
| maximal subgroup of | #107603, #107620, #107851, … |
| maximal normal subgroup of | #107603, #107620, #107851, … |
| maximal quotient of | C2×#7594, C2×#7596, C2×#7599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10646);