Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10512 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
172 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,11)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15)(17,18,22,26)(19,27,30,42)(20,31,34,24)(21,35,37,25)(23,36,39,29)(28,44,32,43)(33,41,48,46)(38,45,40,47);(1,3)(2,7)(4,13)(5,14)(6,12)(8,16)(9,11)(10,15)(17,19,22,30)(18,23,26,39)(20,32,34,28)(21,36,37,29)(24,40,31,38)(25,42,35,27)(33,45,48,47)(41,43,46,44);(2,8)(3,12)(5,6)(9,10)(13,14)(15,16)(17,20,21,33,22,34,37,48)(18,24,25,41,26,31,35,46)(19,28,29,45,30,32,36,47)(23,38,27,43,39,40,42,44)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=dh,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,f^b=fh,e^c=egh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #123187, #123188, #123189, … |
| maximal normal subgroup of | #123187, #123188, #123189, … |
| maximal quotient of | #104862, #104869, #104876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3505);