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×#2913 |
| 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)(10,13)(14,18)(15,19)(17,20)(21,22);(7,8,13,19,11,15,10,12)(9,14,20,22,16,21,17,18);(7,9)(8,14)(10,17)(11,16)(12,18)(13,20)(15,21)(19,22);(9,16)(14,21)(17,20)(18,22)88 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2913⋊C3 |
| maximal subgroup of | #146531, #146533, #146549, … |
| maximal normal subgroup of | #146531, #146533, #146549, … |
| maximal quotient of | C2×#17789, C2×#17799, C2×#17812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53095);