Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1995 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
603 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 603 conjugacy classes of subgroups — too many to draw. Only its 247 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) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,15,18,16,19,14,24)(13,20,23,25,21,26,22,17);(4,7)(8,9)(11,13)(12,17,19,25)(14,22)(15,23)(16,21)(18,20,24,26);(3,5,6,10)(4,8,7,9)(13,21)(17,25)(20,26)(22,23)64 conjugacy classes — view the full character table.
| maximal subgroup of | #107297, #107299, #107353, … |
| maximal normal subgroup of | #107297, #107299, #107353, … |
| maximal quotient of | C2×#9872, C2×#9873, #206777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14316);