Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1111 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 231 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:
(10,13)(14,16)(17,18)(19,22)(20,21)(23,24);(2,4)(5,6,7,8)(10,13)(11,15)(18,21)(19,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,23);(9,11)(10,14)(12,15)(13,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|b^2=h,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #114529, #114531, #114954, … |
| maximal normal subgroup of | #114529, #114531, #114954, … |
| maximal quotient of | #154049, #154050, #154053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7647);