Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C23 |
| commutator, socle | C22 | C22×C42 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C24 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24);(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(10,15)(14,20)(16,22)(21,24);(2,5)(6,8)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=g,f^2=h,c^a=ch,d^a=dg112 conjugacy classes — view the full character table.
| maximal subgroup of | #105319, #105320, #105323, … |
| maximal normal subgroup of | #105319, #105320, #105323, … |
| maximal quotient of | #180800, #180813, #181423, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10311);