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 | #3060 | 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)(11,21)(12,23)(13,24)(14,20)(15,19)(16,22)(17,18);(1,3,4,7)(2,5,6,8)(9,11,15,23,14,18,13,22)(10,16,24,17,20,12,19,21);(2,6)(5,8)(9,11)(10,17)(12,24)(13,22)(14,18)(15,23)(16,19)(20,21);(9,12,15,21,14,16,13,17)(10,18,24,22,20,11,19,23)Polycyclic presentation:
2^8|b^2=ef,d^2=eh,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×#25055, #6444122, #6444124, … |
| maximal normal subgroup of | C2×#25055, #6444122, #6444124, … |
| maximal quotient of | #281069, #281070, #281073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18043);