Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| 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:
(1,2,7,13,15,18,6,11,9,14,24,17,8,3,5,12)(4,19,21,10,27,28,20,16,23,32,25,30,22,31,26,29);(1,3,15,14,9,18,8,2)(4,16,22,30,23,29,27,10)(5,12,7,17,6,11,24,13)(19,21,28,26,32,25,31,20);(1,4)(2,10)(3,16)(5,21)(6,25)(7,26)(8,27)(9,23)(11,31)(12,28)(13,19)(14,30)(15,22)(17,32)(18,29)(20,24)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,e^a=eh,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=egh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13032, #494550, #495737, … |
| maximal normal subgroup of | C2×#13032, #494550, #495737, … |
| maximal quotient of | #136700, #136734, #136909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6020);