Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
357 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24);(2,5)(6,7)(9,10,12,14,13,16,19,21)(11,15,18,20,17,22,23,24);(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,22)(18,23)(20,24)Polycyclic presentation:
2^8|b^2=eh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #263062, #263066, #263102, … |
| maximal normal subgroup of | #263062, #263066, #263102, … |
| maximal quotient of | #189573, #197761, #197765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12184);