Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,12,16,13,14,19,21)(11,15,17,22,18,20,23,24);(2,4)(5,6,8,7);(5,7,8,6)(10,14)(15,20)(16,21)(22,24);(9,11,12,17,13,18,19,23)(10,15,16,22,14,20,21,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,b^a=be,c^a=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | #493218, #493220, #493221, … |
| maximal normal subgroup of | #493218, #493220, #493221, … |
| maximal quotient of | #183846, #183853, #184049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13472);