Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | #432 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(1,2)(3,4)(5,6,8,14,9,12,17,19,10,11,16,18,7,13,15,20);(6,11)(8,16)(12,13)(15,17);(2,4)(6,12,11,13)(8,16)(14,20,18,19)(15,17)Polycyclic presentation:
2^8|a^2=f,c^2=d,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93292, #93293, #93296, … |
| maximal normal subgroup of | #93292, #93293, #93296, … |
| maximal quotient of | #135157, #135188, #135202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5837);