Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9758 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
187 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,28,31,39)(20,32)(21,26)(22,37)(23,27)(24,30,40,36)(25,34)(29,38,44,48)(33,41)(35,46)(42,47,45,43);(1,3,6,11)(2,7)(4,12,5,13)(8,10)(9,15)(14,16)(17,19,21,36,23,31,22,30)(18,24,26,39,27,40,37,28)(20,29,33,47,35,44,34,43)(25,38,32,45,41,48,46,42);(3,11)(7,15)(10,16)(12,13)(17,20,22,34,23,35,21,33)(18,25,26,32,27,41,37,46)(19,29,30,43,31,44,36,47)(24,38,39,45,40,48,28,42)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=e,d^2=gh,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,f^b=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139110, #139114, #139115, … |
| maximal normal subgroup of | #139110, #139114, #139115, … |
| maximal quotient of | #102039, #102042, #102043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2753);