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 | C2×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
328 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 328 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,10,6,9,13,16)(3,8,11,15,4,7,12,14)(17,18)(19,27)(20,31,33,24)(21,34)(22,26)(23,29)(25,37)(28,43,45,48)(30,41)(32,40,36,42)(35,39)(38,44,47,46);(1,3,6,4)(2,7,9,8)(5,12,13,11)(10,15,16,14)(17,19,22,30)(18,23,26,39)(20,28,33,45)(21,35,37,29)(24,38,31,47)(25,41,34,27)(32,46,36,44)(40,43,42,48);(1,4,6,3)(2,8,9,7)(5,11,13,12)(10,14,16,15)(17,20,22,33)(18,24,26,31)(19,28,30,45)(21,36,37,32)(23,38,39,47)(25,42,34,40)(27,43,41,48)(29,46,35,44)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,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 | C2×#8478, #187106, #187110, … |
| maximal normal subgroup of | C2×#8478, #187106, #187110, … |
| maximal quotient of | #97361, #97364, #97367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1466);