Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10488 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #394 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(17,18,22,26)(19,27,28,31)(20,23,21,25)(24,29,32,30);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(18,23)(19,28)(20,21)(24,27)(25,26)(31,32);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,19,20,29,22,28,21,30)(18,24,25,31,26,32,23,27)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,e^b=eh,e^c=egh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10495, #240556, #240560, … |
| maximal normal subgroup of | C2×#10495, #240556, #240560, … |
| maximal quotient of | #104909, #104910, #104915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3483);