Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #121 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #388 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(2,6)(3,10)(4,7)(8,16)(9,14)(11,13)(17,18,20,23,22,26,21,25)(19,27,29,32,28,31,30,24);(1,2,7,14,5,9,4,6)(3,11,15,13,10,8,12,16)(18,23,26,25)(19,28)(24,31,32,27)(29,30);(1,3,4,12,5,10,7,15)(2,8,6,13,9,11,14,16)(17,19,20,29,22,28,21,30)(18,24,25,31,26,32,23,27)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^c=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9588, #203061, #203062, … |
| maximal normal subgroup of | C2×#9588, #203061, #203062, … |
| maximal quotient of | #101905, #101906, #101909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2576);