Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
679 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 219 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:
(2,4)(9,10,12,11)(13,14,18,20)(15,17,16,19);(6,8)(13,15,18,16)(14,19,20,17);(1,2)(3,4)(5,6,7,8)(9,11,12,10)(13,16,18,15)(14,19,20,17);(1,2)(3,4)(13,17,18,19)(14,15,20,16)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=gh,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #140063, C2×#14574, #521053, … |
| maximal normal subgroup of | #140063, C2×#14574, #521053, … |
| maximal quotient of | #153978, #153984, #153999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7562);