Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3555 conjugacy classes of subgroups; only the 2927 normal subgroups are shown.
This group has 3555 conjugacy classes of subgroups — too many to draw. Only its 2927 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(3,4);(1,2)(3,4);(7,11)(9,12)(13,14,17,20)(15,18,23,21)(16,24,25,19)(22,27,28,26);(5,6,8,10)(7,9,11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28);(5,7)(6,9)(8,11)(10,12)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#541 |
| maximal subgroup of | #184772, #184810, #184838, … |
| maximal normal subgroup of | #184772, #184810, #184838, … |
| maximal quotient of | C22×#2320, C22×#2314, C22×#2334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55669);