Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1853 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 1853 conjugacy classes of subgroups — too many to draw. Only its 183 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)(5,6,10,9)(7,11,15,14)(8,12,17,16)(13,19,20,18);(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20);(5,8)(6,12)(7,13)(9,16)(10,17)(11,18)(14,19)(15,20);(2,4)(5,9,10,6)(7,14,15,11)(8,16,17,12)(13,19,20,18)Polycyclic presentation:
2^8|a^2=h,d^2=h,b^a=be,c^a=cf,d^a=dg,f^b=fh,e^c=eh49 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25889);