Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,20)(12,21)(13,18)(14,24)(15,19)(16,17)(22,23);(1,2,3,4)(6,8)(9,11,14,17,15,21,13,23)(10,16,18,20,19,22,24,12);(9,12,14,16,15,20,13,22)(10,17,18,21,19,23,24,11);(9,13,15,14)(10,18,19,24)(11,17,21,23)(12,22,20,16)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #140615, #140681, #355555, … |
| maximal normal subgroup of | #140615, #140681, #355555, … |
| maximal quotient of | #204264, #204281, #204298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14219);