Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
817 conjugacy classes of subgroups; only the 379 normal subgroups are shown.
This group has 817 conjugacy classes of subgroups — too many to draw. Only its 379 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) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,8,7)(10,12)(13,14,17,18)(15,16,19,20);(5,6,8,7)(14,18)(16,20);(9,10)(11,12)(13,15,17,19)(14,16,18,20);(1,3,4,2)(5,7,8,6)(13,16)(14,19)(15,18)(17,20)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=e,d^2=eg,b^a=be,c^a=cf,d^b=de100 conjugacy classes — view the full character table.
| Fitting subgroup of | C4⋊(#376⋊C3) |
| maximal subgroup of | #100369, #100377, #100385, … |
| maximal normal subgroup of | #100369, #100377, #100385, … |
| maximal quotient of | C4⋊C4×#95, #153725, #153830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6798);