Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1879 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,7,12,14,17,6,11,9,13,25,26,8,3,5,16)(4,18,21,10,27,28,19,15,23,31,24,30,22,32,20,29);(1,3,14,13,9,17,8,2)(4,15,22,30,23,29,27,10)(5,16,7,26,6,11,25,12)(18,21,28,20,31,24,32,19);(1,4)(2,10)(3,15)(5,24)(6,21)(7,19)(8,27)(9,23)(11,28)(12,31)(13,30)(14,22)(16,32)(17,29)(18,26)(20,25)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=g,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=egh,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13036, #494556, #495737, … |
| maximal normal subgroup of | C2×#13036, #494556, #495737, … |
| maximal quotient of | #136703, #136704, #136737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6024);