Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2217 |
| 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 | |
377 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 135 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)(14,22)(15,21)(19,20);(7,8,13,22,10,14,9,12,11,15,16,20,17,19,18,21);(4,6)76 conjugacy classes — view the full character table.
| maximal subgroup of | #141453, #141454, #141457, … |
| maximal normal subgroup of | #141453, #141454, #141457, … |
| maximal quotient of | C2×#12432, C2×#12453, C2×#12468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26587);