Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #379 |
| prime radical | C2×2−1+4 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
477 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 477 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8)(10,14)(12,19)(15,23)(16,20)(17,21)(22,24);(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,24)(21,23);(9,11,13,18)(10,15,16,24)(12,21,19,17)(14,22,20,23)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eg,d^c=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93678, #136825, #136826, … |
| maximal normal subgroup of | #93678, #136825, #136826, … |
| maximal quotient of | #96939, #96941, #96942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1307);