Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3054 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,14,18,12)(7,15,8,11,16,20,17,19);(5,7,9,8,10,16,18,17)(6,11,12,15,14,19,13,20);(2,4)(7,16)(8,17)(11,19)(15,20);(5,8,9,16,10,17,18,7)(6,11,13,20,14,19,12,15)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124776, #124890, #124891, … |
| maximal normal subgroup of | #124776, #124890, #124891, … |
| maximal quotient of | #210811, #210812, #210813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14539);