Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3253 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
329 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(2,5)(6,7)(9,11,12,19,14,21,13,20)(10,15,22,24,17,23,16,18);(2,6)(4,8)(5,7)(9,10)(11,15)(12,22)(13,16)(14,17)(18,20)(19,24)(21,23);(2,5)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)Polycyclic presentation:
2^8|b^2=egh,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^c=dg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #263423, #263424, #263425, … |
| maximal normal subgroup of | #263423, #263424, #263425, … |
| maximal quotient of | #189604, #197838, #197839, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12327);