Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10212 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
215 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 49 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)(3,10)(4,6)(5,15)(7,9)(8,11)(12,16)(13,14)(17,18,23,28)(19,29,32,43)(20,33,36,41)(21,27,22,26)(24,37,39,31)(25,34,42,35)(30,38,46,40)(44,47,48,45);(1,3,6,12)(2,8,4,14)(5,13,7,11)(9,16,15,10)(17,19,23,32)(18,24,28,39)(20,30,36,46)(21,37,22,31)(25,40,42,38)(26,43,27,29)(33,44,41,48)(34,47,35,45);(1,4,6,2)(3,10,12,16)(5,15,7,9)(8,11,14,13)(17,20,23,36)(18,25,28,42)(19,30,32,46)(21,35,22,34)(24,38,39,40)(26,33,27,41)(29,44,43,48)(31,47,37,45)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,f^b=fgh,d^c=dh,e^c=eh,f^c=fgh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151363, C2×#10217, #232859, … |
| maximal normal subgroup of | #151363, C2×#10217, #232859, … |
| maximal quotient of | #102511, #102518, #102526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3205);