Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C4⋊C8 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,18,14,17,22,24,15,19,21,16,12,23,11,20);(2,5)(6,8)(9,11,12,21,15,22,14,13)(10,16,17,20,19,18,23,24);(1,3,4,7)(2,6,5,8)(9,11,12,21,15,22,14,13)(10,16,17,20,19,18,23,24)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93311, #93312, #93350, … |
| maximal normal subgroup of | #93311, #93312, #93350, … |
| maximal quotient of | #135634, #135638, #135644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5871);