Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2220 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 127 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)(8,12)(9,16)(10,17)(13,18)(14,19)(15,22)(20,21);(7,8,13,22,10,15,9,12,11,14,16,21,17,20,18,19);(4,6)(8,14)(12,19)(15,20)(21,22)52 conjugacy classes — view the full character table.
| maximal subgroup of | #141503, #141505, C4×#2220, … |
| maximal normal subgroup of | #141503, #141505, C4×#2220, … |
| maximal quotient of | C2×#12435, C2×#12456, C2×#12472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26591);