Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #102 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C2 | #1279 |
| prime radical | C23 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,7,12,23,28,8,13,22,29,6,11,4,20)(3,14,16,25,18,31,15,9,19,24,32,26,17,30,21,27);(1,3,6,17,8,19,7,18)(2,9)(4,21,22,32,23,15,5,16)(10,24)(11,25)(12,26)(13,27)(14,29)(20,31)(28,30)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fg,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=ef,f^a=fh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7181, #126850, #126873, … |
| maximal normal subgroup of | C2×#7181, #126850, #126873, … |
| maximal quotient of | #92939, #92940, #92945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 169);