Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,3)(9,10,11,16,19,23,24,21,13,15,20,22,12,17,18,14);(5,6,7,8)(10,14)(11,18)(12,19)(15,21)(16,17)(20,24)(22,23);(1,3,4,2)(10,15)(14,21)(16,22)(17,23)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=deg,d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ch,d^b=dg,e^b=eg,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94059, #129165, #129177, … |
| maximal normal subgroup of | #94059, #129165, #129177, … |
| maximal quotient of | #129099, #129114, #129131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5662);