Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1553 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
627 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 627 conjugacy classes of subgroups — too many to draw. Only its 199 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 — 4 generators on 26 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12,15,18,16,19,14,23)(13,20,21,25,22,17,24,26);(4,7)(5,10)(8,9)(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,20)(23,26);(4,8)(7,9)(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,20,26)64 conjugacy classes — view the full character table.
| maximal subgroup of | #107457, #107614, #107639, … |
| maximal normal subgroup of | #107457, #107614, #107639, … |
| maximal quotient of | C2×#7580, C2×#7582, #160502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10699);