Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#1296 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
439 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 439 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) = 256 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(3,4,5,6)(8,12)(9,10)(13,14)(15,16)(17,19)(18,21)(20,22);(7,8,10,13,11,14,9,12)(16,19)(17,22)(20,21)76 conjugacy classes — view the full character table.
| maximal subgroup of | #94324, C4×#1296, #105449, … |
| maximal normal subgroup of | #94324, C4×#1296, #105449, … |
| maximal quotient of | C2×#7040, #95227, #95229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1025);