Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13472 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2067 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
158 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,24,7,13,21,12)(3,14,19,32,20,29,31,28)(4,9,15,10,5,22,23,11)(8,25,30,18,27,16,26,17);(1,3,15,25,6,19,5,18,7,20,23,16,21,31,4,17)(2,8,9,28,12,26,11,29,13,27,22,32,24,30,10,14);(2,9)(3,16)(4,15)(5,23)(6,21)(8,26)(10,12)(11,24)(13,22)(14,29)(17,31)(18,19)(20,25)(27,30)Polycyclic presentation:
2^8|a^2=de,b^2=dg,d^2=gh,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,f^b=fh,d^c=dg,e^c=eg,f^c=fg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13472, #499319, #499514, … |
| maximal normal subgroup of | C2×#13472, #499319, #499514, … |
| maximal quotient of | #145904, #145915, #145924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6460);