Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#1279 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(3,4)(5,6)(8,12)(9,10)(13,14)(15,16,18,21,19,22,17,20);(4,6)(7,8,10,13,11,14,9,12)(16,20,22,21)112 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#340, #105283, #105284, … |
| maximal normal subgroup of | C8×#340, #105283, #105284, … |
| maximal quotient of | C2×#7015, #94881, #94885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 877);