Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #388 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,10,13,11,14,9,16)(7,15,12,19,18,20,17,8);(1,2,3,4)(5,7,9,17,11,18,10,12)(6,8,13,15,14,19,16,20);(5,8)(6,12)(7,16)(9,20)(10,15)(11,19)(13,18)(14,17)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11146, #357847, #358412, … |
| maximal normal subgroup of | C2×#11146, #357847, #358412, … |
| maximal quotient of | #116997, #116998, #117037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4134);