Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,14,11,15);(1,3)(2,5)(4,8)(6,7);(10,14)(15,16);(9,11,13,12)(10,15,14,16)Polycyclic presentation:
2^8|a^2=f,d^2=h,f^2=h,b^a=be,c^a=ch,e^a=eg,f^b=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105580, #105581, #105594, … |
| maximal normal subgroup of | #105580, #105581, #105594, … |
| maximal quotient of | #160708, #160709, #160714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10350);