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 | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
287 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,4,6)(3,7,8,5)(10,15)(11,13)(12,19)(16,24)(17,23)(18,21)(20,22);(1,3,4,8)(2,5,6,7);(9,10)(11,16)(12,20)(13,22)(14,17)(15,19)(18,23)(21,24);(9,11,14,13)(10,16,17,22)(12,18,21,19)(15,20,23,24)Polycyclic presentation:
2^8|a^2=e,b^2=e,d^2=h,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 | #136955, #136956, #136959, … |
| maximal normal subgroup of | #136955, #136956, #136959, … |
| maximal quotient of | #493762, #494285, #494788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26617);