Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | Aut(G) = C23×#7737411 (order 4096) · Inn(G) = #386 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
228 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,3,6,8)(9,10)(11,20)(12,24,27,34)(13,28)(14,18)(15,19)(16,22)(17,26,33,25)(21,36,38,30)(23,31)(29,32)(35,37,40,39);(1,3,5,8,4,2,6,7)(9,11,13,22,15,23,14,29)(10,16,18,20,19,32,28,31)(12,21,26,39,27,38,25,37)(17,30,34,35,33,36,24,40);(2,3)(7,8)(9,12,15,27)(10,17,19,33)(11,21,23,38)(13,26,14,25)(16,30,32,36)(18,34,28,24)(20,35,31,40)(22,39,29,37)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,e^b=eg,d^c=dg,e^c=eg,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9092, #196087, #196088, … |
| maximal normal subgroup of | C2×#9092, #196087, #196088, … |
| maximal quotient of | #99804, #99806, #99812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2080);