Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#2218 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
393 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 393 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)(9,13)(10,16)(14,18)(15,22)(17,21)(19,20);(7,8,13,20,10,15,17,18,11,14,21,22,16,19,9,12);(4,6)(8,14)(12,18)(15,19)(20,22)52 conjugacy classes — view the full character table.
| maximal subgroup of | #141504, C2×#7422, C2×#7424, … |
| maximal normal subgroup of | #141504, C2×#7422, C2×#7424, … |
| maximal quotient of | C2×#12433, C2×#12454, C2×#12471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26589);