Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,21)(13,15)(16,18)(17,19)(20,22)(23,24);(1,3,4,8)(2,5,6,7);(10,14)(11,18)(12,19)(15,22)(16,17)(20,24)(21,23);(10,15)(14,22)(16,23)(17,21)Polycyclic presentation:
2^8|a^2=e,b^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136953, #136957, #136971, … |
| maximal normal subgroup of | #136953, #136957, #136971, … |
| maximal quotient of | #493760, #494283, #494787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26615);