Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C25 |
| direct product | C2×#1374 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C22 |
| commutator | C2 | C2×C4×C16 |
| Frattini, cosocle | C22×C8 | C23 |
| socle, prime radical | C24 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,6)(11,12,13,16,14,17,19,22,15,18,20,23,21,24,25,26);(4,6)(7,8,9,10)160 conjugacy classes — view the full character table.
| maximal subgroup of | #93278, #93279, #93329, … |
| maximal normal subgroup of | #93278, #93279, #93329, … |
| maximal quotient of | C2×#7136, #126560, #126572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5531);