Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2971 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 2971 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) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,15)(11,13,16,14);(10,13)(11,16)(14,15);(2,4);(6,8)Polycyclic presentation:
2^8|a^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #32881⋊C3, #72241506, #72252247 |
| maximal subgroup of | #246002, #246006, #246042, … |
| maximal normal subgroup of | #246002, #246006, #246042, … |
| maximal quotient of | #346307, #346311, #346315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25869);