Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
461 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 461 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,19)(17,27)(18,22)(20,25)(21,24)(26,28);(6,9)(10,12)(13,15,24,26,18,25,17,16)(14,19,27,20,22,28,21,23);(1,2,3,4)(13,15,17,16,18,25,24,26)(14,19,21,23,22,28,27,20);(5,7,8,11)(6,10,9,12)(13,16,17,25,18,26,24,15)(14,20,21,19,22,23,27,28)Polycyclic presentation:
2^8|b^2=gh,c^2=f,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=ch,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125419, #125539, #144105, … |
| maximal normal subgroup of | #125419, #125539, #144105, … |
| maximal quotient of | #352679, #352682, #352683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26241);