Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
527 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)(10,15)(11,18)(12,13)(16,23)(17,21)(19,20);(5,6,7,8)(9,10)(11,16)(12,21)(13,15)(14,17)(18,22)(19,24)(20,23);(2,4)(9,11,12,19,14,20,13,18)(10,16,21,24,17,23,15,22)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=efh,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10943, #354109, #354110, … |
| maximal normal subgroup of | C2×#10943, #354109, #354110, … |
| maximal quotient of | #110117, #110120, #110123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3931);