Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #384 |
| prime radical | C23 | C4⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(1,3,4,12,6,11,5,13)(2,7,8,10,9,15,14,16)(18,22)(23,30)(24,29)(25,31);(3,11)(7,15)(10,16)(12,13)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31)Polycyclic presentation:
2^8|a^2=f,b^2=dh,c^2=e,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg52 conjugacy classes — view the full character table.
| maximal subgroup of | #146092, #146093, #146095, … |
| maximal normal subgroup of | #146092, #146093, #146095, … |
| maximal quotient of | #99302, #99303, #99304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1815);