Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C22 | C2×C4×C8 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
401 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 191 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 — 3 generators on 20 points:
(5,6,7,10,8,9,11,12)(13,14,16,19,17,20,15,18);(1,2,3,4)(6,9)(7,11);(13,15,17,16)(14,18,20,19)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=gh,e^2=h,b^a=bd,d^a=dg,e^b=eg88 conjugacy classes — view the full character table.
| maximal subgroup of | #93426, C8×#343, #105307, … |
| maximal normal subgroup of | #93426, C8×#343, #105307, … |
| maximal quotient of | #94917, #94933, #94945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 896);