Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D4 |
| commutator | C4 | C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle, prime radical | C23 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(2,4)(5,6,8,11,9,12,7,10)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=f,g^2=h,b^a=bc,c^a=cf,e^a=ef112 conjugacy classes — view the full character table.
| maximal subgroup of | #92821, #92822, #92866, … |
| maximal normal subgroup of | #92821, #92822, #92866, … |
| maximal quotient of | #92911, #92912, #92913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 139);