Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,11,14,12,15,16,13);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(14,15)(17,18)(19,21)(20,23)(22,24);(10,13)(11,16)(14,15)(18,21)(19,24)(22,23)Polycyclic presentation:
2^8|a^2=d,b^2=deg,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=eg,f^b=fh,d^c=dg,e^c=eg,f^c=fh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #10673⋊C3 |
| maximal subgroup of | #94716, #126279, #126337, … |
| maximal normal subgroup of | #94716, #126279, #126337, … |
| maximal quotient of | #105231, #117094, #121392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3661);