Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
309 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,4,3)(5,6,8,11)(7,9,12,10)(14,18)(15,16)(19,20);(6,9,11,10)(13,14,16,19,17,20,15,18);(1,3,4,2)(13,15,17,16)(14,18,20,19)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=eg,d^2=g,f^2=h,b^a=bd,c^a=ch,d^a=dg,f^a=fg88 conjugacy classes — view the full character table.
| maximal subgroup of | #98750, #98751, #98754, … |
| maximal normal subgroup of | #98750, #98751, #98754, … |
| maximal quotient of | #96709, #96713, #96745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1165);