Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 451 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,21)(19,23)(22,24);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,18)(19,24)(20,22)(21,23);(18,22)(21,24)Polycyclic presentation:
2^8|a^2=e,b^2=ef,c^2=e,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=dh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101272, #101273, #101276, … |
| maximal normal subgroup of | #101272, #101273, #101276, … |
| maximal quotient of | #244085, #244088, #244092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16733);