Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
362 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,19)(13,17)(14,18)(16,23)(20,22)(21,24);(1,3,4,7)(2,5,6,8)(9,11,13,19,14,21,23,20)(10,12,16,15,18,22,17,24);(2,6)(5,8)(9,11)(10,15)(12,17)(13,19)(14,21)(16,22)(18,24)(20,23);(9,12,13,15,14,22,23,24)(10,11,16,19,18,21,17,20)Polycyclic presentation:
2^8|b^2=ef,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25054, #6444122, #6444123, … |
| maximal normal subgroup of | C2×#25054, #6444122, #6444123, … |
| maximal quotient of | #281067, #281068, #281071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18042);