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) = C2×C4 |
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 | C23 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3)(5,6,9,15,8,14,17,11,10,13,19,20,18,12,7,16);(1,2,4,3)(5,7,8,9,10,17,18,19)(6,11,14,20,13,16,12,15);(1,3,4,2)(6,12,13,14)(7,17)(9,19)(11,20,16,15)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=d,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=ch,f^b=fh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93608, #93613, #124541, … |
| maximal normal subgroup of | #93608, #93613, #124541, … |
| maximal quotient of | #135633, #135635, #135636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5870);