Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 4096 |
| 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 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,10,12,16)(11,13,18,19);(5,6,8,16,9,12,17,10)(7,13,15,11,14,19,20,18);(2,4)(7,14)(11,18)(13,19)(15,20);(5,7,8,15,9,14,17,20)(6,11,10,13,12,18,16,19)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,e^2=h,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #504520, #504521, #504522, … |
| maximal normal subgroup of | #504520, #504521, #504522, … |
| maximal quotient of | #229114, #229115, #229116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16008);