Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1879 · Out(G) = C24 |
| 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 | #3414 | 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:
(1,2,5,10,6,12,22,26,7,13,4,11,21,30,19,8)(3,14,16,27,17,29,23,28,18,25,15,32,24,31,20,9);(2,8,13,26)(4,19)(5,22)(6,21)(9,25,28,14)(10,12,11,30)(15,20)(16,23)(17,24)(27,29,32,31);(1,3)(2,9)(4,20)(5,23)(6,24)(7,18)(8,25)(10,29)(11,31)(12,32)(13,28)(14,26)(15,19)(16,22)(17,21)(27,30)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=g,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×#13030, #494547, #495739, … |
| maximal normal subgroup of | C2×#13030, #494547, #495739, … |
| maximal quotient of | #136699, #136700, #136733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6018);