Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C23 |
| commutator | C22 | C22×C16 |
| socle | C23 | C22×C8 |
| prime radical | C25 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
599 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)(3,4)(5,6)(7,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(2,4);(6,8)Polycyclic presentation:
2^8|a^2=f,f^2=g,g^2=h,b^a=bd,c^a=ce112 conjugacy classes — view the full character table.
| Fitting subgroup of | #12679⋊C3 |
| maximal subgroup of | #93462, #93463, #93517, … |
| maximal normal subgroup of | #93462, #93463, #93517, … |
| maximal quotient of | #129201, #129240, #129275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5667);