Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×#127 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
407 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 147 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:
(2,5)(3,7)(6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(9,11,13,20,14,21,12,19)(10,15,16,18,17,23,22,24);(1,2,3,5,4,6,7,8);(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #122897, #122905, #122909, … |
| maximal normal subgroup of | #122897, #122905, #122909, … |
| maximal quotient of | #191419, #206079, #206085, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12584);