Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10)(11,18)(12,17)(13,21)(14,16)(15,19)(20,24)(22,23);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(1,3,4,8,5,2,6,7)(10,16)(15,23)(17,21)(18,24);(9,12,14,13)(10,17,16,21)(11,19,20,22)(15,24,23,18)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eg88 conjugacy classes — view the full character table.
| maximal subgroup of | #128690, #128691, #128721, … |
| maximal normal subgroup of | #128690, #128691, #128721, … |
| maximal quotient of | #182401, #183430, C4×SD16⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10837);