Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#1330 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C23 | C22⋊C8 |
| prime radical | C22×D4 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
211 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 128 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,13,17,15,18,20,24,14,16,19,22,21,23,25,26);(4,7)(5,10)(8,9)(12,16)(13,19)(18,23)(20,25)76 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1330, #128522, #128523, … |
| maximal normal subgroup of | C4×#1330, #128522, #128523, … |
| maximal quotient of | C2×#7072, C2×#7073, #122488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4573);