Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(6,11,13,19)(7,15)(9,18)(12,20,14,16);(2,4)(5,6,9,14,8,11,15,20,10,13,18,12,17,19,7,16);(5,7,8,9,10,15,17,18)(6,12,19,20,13,14,11,16)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ch,f^a=fh,c^b=cg,e^b=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93617, #93621, #93650, … |
| maximal normal subgroup of | #93617, #93621, #93650, … |
| maximal quotient of | #129054, #129068, #129079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5646);