Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C2×D4 · Out(G) = C210 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
418 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(18,21)(22,23);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,21)(20,22)(23,24);(10,14)(13,16)(18,22)(20,24)(21,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|b^2=f,d^2=eh,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #212160, #212166, #212172, … |
| maximal normal subgroup of | #212160, #212166, #212172, … |
| maximal quotient of | #358131, #358380, #358381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16746);