Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2087 |
| prime radical | C23 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,17,9,28,25,6,11,19,27,5,10,18,26)(3,12,15,21,20,29,32,22,14,30,31,23,13,24,16,7);(3,13,14,20)(7,21,22,23)(12,29,30,24)(15,16,31,32);(1,3,5,20,6,14,17,13)(2,7,10,21,11,22,9,23)(4,16,18,15,19,32,28,31)(8,24,26,12,27,29,25,30)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^c=dh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93852, #93853, #151509, … |
| maximal normal subgroup of | #93852, #93853, #151509, … |
| maximal quotient of | #146151, #146159, #146178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6507);