Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
353 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22);(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24);(2,5)(6,8)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=g,e^2=h,b^a=bd,c^a=ch,f^a=fg,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105582, #105583, #106018, … |
| maximal normal subgroup of | #105582, #105583, #106018, … |
| maximal quotient of | #94864, #94866, #94872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 843);