Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
901 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 901 conjugacy classes of subgroups — too many to draw. Only its 255 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,19)(18,20);(6,9)(10,12);(14,17)(15,20)(18,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=g,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,f^c=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #105688, #105694, #105956, … |
| maximal normal subgroup of | #105688, #105694, #105956, … |
| maximal quotient of | #190666, #190667, #190669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13720);