Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #390 |
| prime radical | C23 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
227 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,17)(12,14)(13,21)(15,19)(16,20)(18,23)(22,24);(2,6,8,7)(10,14)(11,18)(16,21)(19,24);(9,11,13,19,12,18,20,24)(10,15,16,17,14,22,21,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=e,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #92856, #138738, #138742, … |
| maximal normal subgroup of | #92856, #138738, #138742, … |
| maximal quotient of | #102116, #102119, #102120, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2838);