Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
466 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,29)(20,35,37,40)(22,33)(24,32,28,34)(27,31)(30,36,39,38);(2,5)(3,7)(9,11)(10,15)(12,20)(13,27)(14,22)(16,30)(17,33)(18,31)(19,26)(21,29)(23,39)(24,38)(25,37)(28,36)(32,35)(34,40);(2,5)(3,7)(9,12)(10,16)(11,20)(13,28)(14,25)(15,30)(17,34)(18,23)(19,35)(21,38)(22,37)(24,29)(26,32)(27,36)(31,39)(33,40)Polycyclic presentation:
2^8|a^2=f,d^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #152025, #152026, #152328, … |
| maximal normal subgroup of | #152025, #152026, #152328, … |
| maximal quotient of | #97360, #97363, #97366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1463);