Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1391 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 1391 conjugacy classes of subgroups — too many to draw. Only its 607 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,8)(9,10,12,11)(14,17)(15,19);(13,14,16,17)(15,18,19,20);(1,2,3,4)(5,6,7,8)(9,11,12,10)(13,14)(15,18)(16,17)(19,20);(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #99146, #99147, #101055, … |
| maximal normal subgroup of | #99146, #99147, #101055, … |
| maximal quotient of | #153717, #153718, #153852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6783);