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:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17,14,11,18,16)(12,15,21,23,20,19,24,22);(2,5)(3,8)(6,7)(9,11)(10,14)(12,19)(13,16)(15,20)(17,18)(21,22)(23,24);(9,12,13,21,14,20,18,24)(10,15,17,23,11,19,16,22);(2,6)(5,7)(10,11)(12,20)(16,17)(21,24)Polycyclic presentation:
2^8|a^2=f,c^2=f,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×#18076, #6345122, #6345140, … |
| maximal normal subgroup of | C2×#18076, #6345122, #6345140, … |
| maximal quotient of | #183168, #183169, #183688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11064);