Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
797 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 797 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(2,4)(9,10,12,13)(11,15,16,14)(17,18,22,21)(19,23,20,24);(6,8)(10,13)(14,15)(17,19,22,20)(18,23,21,24);(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,20,22,19)(18,23,21,24);(1,2)(3,4)(9,11,12,16)(10,14,13,15)(17,21,22,18)(19,23,20,24)Polycyclic presentation:
2^8|a^2=f,b^2=ef,c^2=ef,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | #144654, C2×#17136, #844904, … |
| maximal normal subgroup of | #144654, C2×#17136, #844904, … |
| maximal quotient of | #156484, #156489, #156542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10124);