Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| 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 | |
163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,14,8,11,17,20,10,15,19,12,18,13,7,16);(1,2,3,4)(6,11,15,13)(12,16,14,20);(5,7,8,9,10,17,18,19)(6,12,13,20,15,14,11,16)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=d,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=cg,d^a=dh,f^a=fh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93354, #93620, #93621, … |
| maximal normal subgroup of | #93354, #93620, #93621, … |
| maximal quotient of | #128212, #128220, #128223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5583);