Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| 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 | C2×#449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
593 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)(10,14)(12,19)(15,21)(16,20)(17,23)(22,24);(5,7,8,6)(9,10)(11,15)(12,20)(13,16)(14,19)(17,24)(18,22)(21,23);(2,4)(9,11,13,18)(10,15,16,22)(12,17,19,23)(14,21,20,24);(10,14)(12,19)(15,21)(16,20)(17,23)(22,24)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=deh,e^b=eh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22872, #6381388, #6381399, … |
| maximal normal subgroup of | C2×#22872, #6381388, #6381399, … |
| maximal quotient of | #227000, #227007, #227013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15860);