Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| 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 | #3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
452 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)(11,20)(12,17)(13,22)(14,21)(15,18)(16,19)(23,24);(1,3,4,7)(2,5,6,8)(9,11,16,23)(10,13,19,12)(14,17,15,22)(18,24,21,20);(2,6)(5,8)(9,12,16,13)(10,17,19,22)(11,21,23,18)(14,24,15,20);(9,13,16,12)(10,11,19,23)(14,24,15,20)(17,21,22,18)Polycyclic presentation:
2^8|b^2=fh,c^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24932, #6438378, #6438380, … |
| maximal normal subgroup of | C2×#24932, #6438378, #6438380, … |
| maximal quotient of | #279840, #279841, #279844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17920);