Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C25×#120 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
218 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,5,9,6,10,4,8)(3,11,13,7,14,15,12,16)(18,22,24,27)(19,25,26,28);(1,3)(2,7)(4,13)(5,12)(6,14)(8,15)(9,11)(10,16)(17,18,20,27,21,24,29,22)(19,23,28,31,26,32,25,30);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,15)(10,16)(17,19)(18,23)(20,28)(21,26)(22,30)(24,32)(25,29)(27,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=g,e^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=egh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=eg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9051, #195424, #195425, … |
| maximal normal subgroup of | C2×#9051, #195424, #195425, … |
| maximal quotient of | #99579, #99580, #99583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2039);