Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×H(3,𝔽4) | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
575 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 575 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) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7)(5,9)(8,10)(11,12,16,20)(13,21,23,26)(14,19,15,18)(17,24,25,22);(5,9)(7,10)(11,13)(12,17)(14,24)(15,22)(16,23)(18,26)(19,21)(20,25);(3,4)(5,10)(6,8)(7,9)(11,13)(12,17)(14,22)(15,24)(16,23)(18,21)(19,26)(20,25)40 conjugacy classes — view the full character table.
| maximal subgroup of | #152576, #153161, #153165, … |
| maximal normal subgroup of | #152576, #153161, #153165, … |
| maximal quotient of | C2×#7750, C2×#7786, C2×#7758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11967);