Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
359 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)(5,6,9,13,10,15,8,14)(7,12,17,19,18,20,16,11);(5,7,8,16,10,18,9,17)(6,11,13,12,15,19,14,20);(1,2,3,4)(5,7)(6,12)(8,16)(9,17)(10,18)(11,14)(13,19)(15,20)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=e,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11341, #358489, #358519, … |
| maximal normal subgroup of | C2×#11341, #358489, #358519, … |
| maximal quotient of | #121334, #121401, #121402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4329);