Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#2912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1383 conjugacy classes of subgroups; only the 999 normal subgroups are shown.
This group has 1383 conjugacy classes of subgroups — too many to draw. Only its 999 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 — 5 generators on 22 points:
(1,2);(3,4,5,6)(8,12)(10,17)(13,18)(14,19)(16,21)(20,22);(7,8,10,12,11,14,17,19)(9,13,16,18,15,20,21,22);(7,9)(8,13)(10,16)(11,15)(12,18)(14,20)(17,21)(19,22);(9,15)(13,20)(16,21)(18,22)88 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2912⋊C3 |
| maximal subgroup of | #146535, #146537, #146551, … |
| maximal normal subgroup of | #146535, #146537, #146551, … |
| maximal quotient of | C2×#17788, C2×#17798, C2×#17811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53094);