Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
502 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 125 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:
(1,2)(3,4)(9,10,14,18)(11,16,20,24)(12,21,19,15)(13,23,22,17);(5,6)(7,8)(9,11,14,20)(10,15,18,21)(12,22,19,13)(16,17,24,23);(2,4)(6,8);(2,4)(9,12,13,11,14,19,22,20)(10,16,17,21,18,24,23,15)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cg,d^b=de,e^b=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123216, #123232, #6632658, … |
| maximal normal subgroup of | #123216, #123232, #6632658, … |
| maximal quotient of | #339755, #339756, #339759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23840);