Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| 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 | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,14)(13,23)(15,19)(16,20)(17,18)(22,24);(1,3,5,4)(2,6,7,8)(9,11,14,18,16,23,15,22)(10,17,19,21,20,24,12,13);(1,4,5,3)(2,6,7,8)(9,12,14,10,16,19,15,20)(11,17,18,21,23,24,22,13);(2,7)(6,8)(9,13,14,17,16,21,15,24)(10,18,19,23,20,22,12,11)Polycyclic presentation:
2^8|b^2=e,c^2=e,d^2=fh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26849, #6443957, #6443963, … |
| maximal normal subgroup of | C2×#26849, #6443957, #6443963, … |
| maximal quotient of | #311002, #311003, #311005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19837);