Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×Q8 | C23 |
| socle | C22 | C2×#99 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,14,10,15,7,12)(8,13,16,20,17,11,18,19);(1,2,3,4)(6,11,15,13)(12,19,14,20);(5,7,10,9)(6,12,15,14)(8,16,17,18)(11,19,13,20)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=h,e^2=h,b^a=bd,d^a=dg,f^a=fh,g^a=gh,e^b=egh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94278, #94282, #94310, … |
| maximal normal subgroup of | #94278, #94282, #94310, … |
| maximal quotient of | #126914, #126951, #126978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5573);