Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,14,17,15,18,12,13)(11,19,20,23,21,24,22,16);(5,7,8,6)(9,11,15,21)(10,16,18,23)(12,20,14,22)(13,19,17,24);(2,4)(9,12,15,14)(10,13,18,17)(11,20,21,22)(16,19,23,24);(9,13,12,18,15,17,14,10)(11,16,22,24,21,23,20,19)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=h,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #498131, #498134, #498139, … |
| maximal normal subgroup of | #498131, #498134, #498139, … |
| maximal quotient of | #229106, #229124, #229144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16014);