Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
848 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 848 conjugacy classes of subgroups — too many to draw. Only its 145 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:
(5,6,12,16)(7,17,9,20)(8,19,13,18)(10,15,11,14);(1,2)(3,4)(5,7)(6,8)(9,12)(10,19)(11,18)(13,16)(14,20)(15,17);(2,4)(5,8,12,13)(6,9,16,7)(10,17,11,20)(14,19,15,18);(5,9)(6,13)(7,12)(8,16)(10,18)(11,19)(14,17)(15,20)Polycyclic presentation:
2^8|a^2=h,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,e^b=eh,f^c=fh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #273332, #273334, #273368, … |
| maximal normal subgroup of | #273332, #273334, #273368, … |
| maximal quotient of | #244660, #244670, #244675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16868);