Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
461 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 461 conjugacy classes of subgroups — too many to draw. Only its 287 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,16,18,20,25,24)(15,19,23,21,22,26,28,27);(1,2,3,4)(6,9)(10,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28);(13,16,17,20,18,24,25,14)(15,21,23,26,22,27,28,19);(5,7,8,11)(6,10,9,12)(15,22)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=e,f^2=h,b^a=be,c^a=ch,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #109868, #109871, #109937, … |
| maximal normal subgroup of | #109868, #109871, #109937, … |
| maximal quotient of | #183727, #183735, #183744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13465);