Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,18,14,17,12,16)(11,15,19,22,20,23,21,24);(9,11,13,21,14,20,12,19)(10,15,16,22,17,23,18,24);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,19,20,21)(15,22,23,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=h,d^2=f,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #106082, #107156, #107171, … |
| maximal normal subgroup of | #106082, #107156, #107171, … |
| maximal quotient of | #207662, #207663, #207666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14414);