Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#1888 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
939 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 939 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) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(7,8,13,16,12,9,11,15)(10,14,20,22,19,17,18,21);(3,4,5,6)(7,9,12,8)(10,17,19,14)(11,15,13,16)(18,21,20,22);(7,10,12,19)(8,14,9,17)(11,20,13,18)(15,22,16,21)64 conjugacy classes — view the full character table.
| maximal subgroup of | #146629, #146631, #146648, … |
| maximal normal subgroup of | #146629, #146631, #146648, … |
| maximal quotient of | C2×#8960, C2×#8964, #193809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13939);