Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1533 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
655 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 655 conjugacy classes of subgroups — too many to draw. Only its 191 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);(3,4,5,8,6,7,9,10)(11,12)(13,19,21,26)(14,18)(15,17)(16,20,24,22)(23,25);(4,7)(5,9)(11,13)(12,16)(14,22)(15,21)(17,24)(18,26)(19,25)(20,23);(12,17)(16,24)(18,25)(19,26)52 conjugacy classes — view the full character table.
| maximal subgroup of | #106070, #107287, #107760, … |
| maximal normal subgroup of | #106070, #107287, #107760, … |
| maximal quotient of | C2×#7592, C2×#7595, C2×#7601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10644);