Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
341 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(2,4)(5,6)(7,11,12,9)(8,10)(13,14,15,17,16,18,19,20);(1,2,3,4);(5,7,8,12)(6,9,10,11)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,f^c=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93336, #93503, #93594, … |
| maximal normal subgroup of | #93336, #93503, #93594, … |
| maximal quotient of | #98781, #98784, #98786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1546);