Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,22)(13,18)(14,19)(16,21)(17,23)(20,24);(1,3,4,8)(2,5,6,7)(9,11,14,21)(10,15,19,16)(12,23,20,13)(17,24,18,22);(9,12,23,11,14,20,13,21)(10,16,18,24,19,15,17,22);(2,6)(5,7)Polycyclic presentation:
2^8|b^2=eh,c^2=fh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25526, #6452220, #6452227, … |
| maximal normal subgroup of | C2×#25526, #6452220, #6452227, … |
| maximal quotient of | #290821, #290822, #290823, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18514);