Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
999 conjugacy classes of subgroups; only the 311 normal subgroups are shown.
This group has 999 conjugacy classes of subgroups — too many to draw. Only its 311 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:
(1,2,4,3)(5,6,8,7)(10,13)(11,15)(17,18,20,21)(19,22,23,24);(5,6,8,7)(10,13)(14,16)(18,21)(19,23);(1,3,4,2)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24);(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=eg,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de76 conjugacy classes — view the full character table.
| maximal subgroup of | #100771, #100775, #100778, … |
| maximal normal subgroup of | #100771, #100775, #100778, … |
| maximal quotient of | #153730, #153731, #153970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6820);