Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | Aut(G) order 16384 |
| 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 | #3492 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
543 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 543 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,18)(12,16)(13,15)(14,19)(17,21)(20,24)(22,23);(1,3,5,4)(2,6,7,8)(9,11,12,19)(10,14,16,18)(13,22,21,20)(15,24,17,23);(1,4,5,3)(2,6,7,8)(10,15)(13,21)(14,23)(16,17)(18,24)(20,22);(2,7)(6,8)(10,16)(14,18)(15,17)(23,24)Polycyclic presentation:
2^8|b^2=e,c^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139639, #139641, #139657, … |
| maximal normal subgroup of | #139639, #139641, #139657, … |
| maximal quotient of | #256292, #256294, #256300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17065);