Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
453 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,13,10,15,25,14)(7,16,18,29,19,33,24,32)(8,20,22,12,23,30,36,31)(11,21,26,17,28,34,27,35);(5,7,10,19)(6,11,15,28)(8,21,23,34)(9,24,25,18)(12,29,31,32)(13,27,14,26)(16,20,33,30)(17,22,35,36);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dgh,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130723, #130724, #130731, … |
| maximal normal subgroup of | #130723, #130724, #130731, … |
| maximal quotient of | #129504, #129556, #129765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5725);