Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 151 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:
(1,2,4,6)(3,7,8,5)(10,14)(11,18)(12,15)(17,22)(19,20)(21,23);(1,3,4,8)(2,5,6,7);(9,10,15,22,13,17,12,14)(11,16,18,23,20,24,19,21);(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,24)(22,23)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=fh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #138108, #138109, #138120, … |
| maximal normal subgroup of | #138108, #138109, #138120, … |
| maximal quotient of | #352404, #352405, #352410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26210);