Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 185 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,22,20,23,19,24);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,18,19,24,20,23,21,22);(9,12,13,11)(10,14,15,16)(17,19,20,21)(18,22,23,24);(1,3,4,7)(2,6,5,8)(17,19,20,21)(18,22,23,24)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #99400, #99401, #99402, … |
| maximal normal subgroup of | #99400, #99401, #99402, … |
| maximal quotient of | #356337, #356910, #357273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16237);