Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3171 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,14,17,15,11,13,19)(12,16,20,24,22,18,21,23);(1,2,5,7)(3,6,4,8)(9,11,15,10)(12,18,22,16)(13,17,14,19)(20,23,21,24);(9,12,14,20,15,22,13,21)(10,16,17,24,11,18,19,23);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,17,11,19)(12,20,22,21)(16,23,18,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,d^a=dg,e^a=eg,d^b=dh,e^b=eg,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18078, #6345122, #6345140, … |
| maximal normal subgroup of | C2×#18078, #6345122, #6345140, … |
| maximal quotient of | #183168, #183171, #183687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11066);