Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2073 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
639 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 639 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) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(16,19)(20,22);(8,11)(12,14)(16,20)(18,21)(19,22);(4,6)(7,9,10,13)(8,12,11,14)(15,16)(17,19)(18,22)(20,21)64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7412, #150587, C4×#2073, … |
| maximal normal subgroup of | C2×#7412, #150587, C4×#2073, … |
| maximal quotient of | C2×#10621, C2×#10625, C2×#10622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16612);