Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
541 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 151 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 — 4 generators on 24 points:
(2,3)(5,6,7,8)(9,10,14,16,12,17,13,18)(11,19,22,23,21,15,20,24);(1,2,4,3)(9,11,13,20,12,21,14,22)(10,15,18,23,17,19,16,24);(6,8)(10,16)(11,20)(13,14)(17,18)(21,22)(23,24);(1,3,4,2)Polycyclic presentation:
2^8|a^2=fh,b^2=eg,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #288756, #288768, #288790, … |
| maximal normal subgroup of | #288756, #288768, #288790, … |
| maximal quotient of | #321307, #321313, #321324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21459);