Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
567 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)(4,5)(6,8)(9,10)(11,17)(12,15)(13,22)(14,19)(16,21)(18,23)(20,24);(2,5)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,23)(20,21)(22,24);(11,18,19,20)(14,23,17,24);(1,3,4,8)(2,6,5,7)(10,15)(14,17)(16,22)(23,24)Polycyclic presentation:
2^8|c^2=f,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25592, #6449085, #6449093, … |
| maximal normal subgroup of | C2×#25592, #6449085, #6449093, … |
| maximal quotient of | #293279, #293284, #293289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18580);