Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#33432 (order 512) · Inn(G) = C22 · Out(G) = C2×#504 |
| homology | Abelianization (H1) = C2×C64 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊C64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C32 | C22 |
| commutator | C2 | C2×C64 |
| socle, prime radical | C22 | C2×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 68 points:
(2,4)(5,6,7,12,8,13,17,27,9,14,18,28,21,31,37,47,10,15,19,29,22,32,38,48,24,34,40,50,43,53,57,62,11,16,20,30,23,33,39,49,25,35,41,51,44,54,58,63,26,36,42,52,45,55,59,64,46,56,60,65,61,66,67,68);(1,2,3,4)Polycyclic presentation:
2^8|a^2=d,b^2=c,d^2=e,e^2=f,f^2=g,g^2=h,b^a=bc160 conjugacy classes — view the full character table.
| maximal subgroup of | D4⋊C64, #94604, Q8⋊C64, … |
| maximal normal subgroup of | D4⋊C64, #94604, Q8⋊C64, … |
| maximal quotient of | #94745, #94747, #94755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 531);