Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| 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 | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
647 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 647 conjugacy classes of subgroups — too many to draw. Only its 149 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:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(1,3,4,8)(2,5,6,7);(9,11)(10,14)(12,21)(13,19)(15,24)(16,23)(17,20)(18,22);(12,22)(15,20)(17,24)(18,21)Polycyclic presentation:
2^8|a^2=eh,b^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,f^c=fh,g^c=gh,f^d=fh49 conjugacy classes — view the full character table.
| maximal subgroup of | #144274, #144275, #144288, … |
| maximal normal subgroup of | #144274, #144275, #144288, … |
| maximal quotient of | #349196, #349197, #349201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25959);