Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C2×#117 | C22 |
| socle | C22 | #342 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
172 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,17,14,18,26,33)(11,19,21,34,22,30,35,39)(12,16,24,31,25,32,37,40)(15,27,28,23,29,36,38,20);(2,5,7,6)(9,11)(10,15)(12,23,25,20)(13,21)(14,22)(16,30,32,19)(17,28)(18,29)(24,36,37,27)(26,35)(31,39,40,34)(33,38)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=h,b^a=bc,c^a=cf,e^a=efg,c^b=cg,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7252, #127500, #127537, … |
| maximal normal subgroup of | C2×#7252, #127500, #127537, … |
| maximal quotient of | #93039, #93040, #93043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 240);