Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1857 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
701 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 701 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);(3,4,6,5)(8,12)(10,17)(13,21)(14,18)(15,19)(20,22);(3,5,6,4)(7,8)(9,13)(10,18)(11,14)(12,17)(15,20)(16,22)(19,21);(7,9,11,16)(8,13,14,22)(10,19,17,15)(12,20,18,21)64 conjugacy classes — view the full character table.
| maximal subgroup of | #147343, #147344, #148261, … |
| maximal normal subgroup of | #147343, #147344, #148261, … |
| maximal quotient of | C2×#8836, C2×#8838, #191767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13842);