Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,21)(12,24)(13,18)(14,19)(15,20)(16,22)(17,23);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,13)(14,22)(15,23)(17,19)(18,21)(20,24);(1,3,5,4)(2,6,7,8)(9,12,15,22)(10,17,20,21)(11,13,23,14)(16,19,24,18);(9,13,15,14)(10,18,20,19)(11,12,23,22)(16,21,24,17)Polycyclic presentation:
2^8|c^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,e^b=eg,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #210262, #210270, #210278, … |
| maximal normal subgroup of | #210262, #210270, #210278, … |
| maximal quotient of | #165721, #165760, #165768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11508);