Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C42 | C24 |
| commutator | C2 | C22×C4×C8 |
| socle | C22 | C22×C42 |
| prime radical | C24 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
457 conjugacy classes of subgroups; only the 387 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 387 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:
(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(1,2,3,5,4,6,7,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=g,f^2=h,d^a=dg,c^b=cg136 conjugacy classes — view the full character table.
| maximal subgroup of | #108434, #108436, #108748, … |
| maximal normal subgroup of | #108434, #108436, #108748, … |
| maximal quotient of | #180786, #180788, #181422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10302);