Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 147 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)(6,10)(7,13)(8,16)(12,18)(14,15)(17,20);(5,6,8,10,9,12,16,18)(7,11,14,17,15,19,13,20);(5,7)(6,11)(8,14)(9,15)(10,17)(12,19)(13,16)(18,20);(2,4)(6,12)(10,18)(11,19)(17,20)Polycyclic presentation:
2^8|b^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6470415, #6470416, #6470417, … |
| maximal normal subgroup of | #6470415, #6470416, #6470417, … |
| maximal quotient of | #347758, #347759, #347760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25932);