Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8803 |
| automorphism | Aut(G) = C2×#72260711 (order 3072) · Inn(G) = #383 · Out(G) = C22×A4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #383 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #126 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
108 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)(3,10,12,14)(4,6)(5,9)(7,11,15,16)(8,13)(17,18)(19,27,29,32)(20,24,30,23)(21,25)(22,26)(28,31);(2,6,9,8)(7,10,15,14)(17,19,28,30,22,29,21,20)(18,23,25,27,26,24,31,32);(1,3,4,11,5,12,13,16)(2,7,8,14,9,15,6,10)(17,20,21,29,22,30,28,19)(18,24,25,32,26,23,31,27)Polycyclic presentation:
2^8|a^2=f,b^2=deg,c^2=d,d^2=gh,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fh,f^c=fgh40 conjugacy classes — view the full character table.
| Fitting subgroup of | #8804⋊C3 |
| maximal subgroup of | C2×#8804, #191311, #191312, … |
| maximal normal subgroup of | C2×#8804, #191311, #191312, … |
| maximal quotient of | #99291, #99292, #99293, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1792);