Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
313 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(6,9)(7,10)(11,12)(14,17)(15,19)(18,20);(2,4)(5,6,10,12,8,11,7,9)(14,18)(17,20);(13,14,19,20,16,18,15,17)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eh,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #94804, #131836, #136491, … |
| maximal normal subgroup of | #94804, #131836, #136491, … |
| maximal quotient of | #102122, #102146, #102170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2830);