Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #72317497 (order 2048) · Inn(G) = C2×D4 · Out(G) = C4×#120 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C2×D4 |
| commutator | C4 | C22×C16 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | C2×C4⋊C16 |
| prime radical | C23 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
99 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,6,12,23,34,7,10,24,32,26,33,49,55,8,13,25,35,27,36,50,56,4,20,21,46,22,47,48,62)(3,14,17,38,18,39,43,59,19,29,44,52,45,53,61,63,15,37,40,57,41,58,60,64,16,9,28,30,42,31,51,54);(1,3,4,16,8,15,7,19)(2,9,10,14,13,29,20,37)(5,17,21,28,25,40,24,44)(6,18,22,42,27,41,26,45)(11,30,32,38,35,52,46,57)(12,31,33,39,36,53,47,58)(23,43,48,51,50,60,49,61)(34,54,55,59,56,63,62,64);(3,15)(9,29)(14,37)(16,19)(17,40)(18,41)(28,44)(30,52)(31,53)(38,57)(39,58)(42,45)(43,60)(51,61)(54,63)(59,64)Polycyclic presentation:
2^8|a^2=e,b^2=d,d^2=h,e^2=f,f^2=g,g^2=h,b^a=bd,d^a=dh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #151015, #151016, #151017, … |
| maximal normal subgroup of | #151015, #151016, #151017, … |
| maximal quotient of | #151020, #151027, #151043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6618);