Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(9,10,11,12)(14,15)(16,17)(19,20);(5,6)(7,8)(13,14,16,15,18,20,17,19);(2,4)(6,8)(13,15,16,20,18,19,17,14)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf64 conjugacy classes — view the full character table.
| maximal subgroup of | #93049, #93050, #131369, … |
| maximal normal subgroup of | #93049, #93050, #131369, … |
| maximal quotient of | #120166, #120179, #120184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4219);