Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) = C2×#72215179 (order 2048) · Inn(G) = #120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | C23 | #123 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,7,11,6,9)(3,12,8,15,14,16,13,4)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(1,3,5,8,7,14,6,13)(2,4,10,12,11,15,9,16)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,4,7,15)(2,8,11,13)(3,10,14,9)(5,12,6,16)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=g,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dg,e^b=eh,d^c=dh,e^c=egh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94602, C2×#8919, #193148, … |
| maximal normal subgroup of | #94602, C2×#8919, #193148, … |
| maximal quotient of | #99327, #99328, #99330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1907);