Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 689 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,15)(11,17,21,24)(12,22,18,16)(14,23,19,20);(9,11,18,23,13,21,12,20)(10,14,22,24,15,19,16,17);(2,4)(9,10)(11,19)(12,16)(13,15)(14,21)(17,23)(18,22)(20,24);(6,8)(10,15)(14,19)(16,22)(17,24)Polycyclic presentation:
2^8|a^2=h,b^2=eh,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,c^b=ch,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137969, #137970, #137985, … |
| maximal normal subgroup of | #137969, #137970, #137985, … |
| maximal quotient of | #347734, #347735, #347736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25928);