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 | #1277 |
| prime radical | C23 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,11,7,13,22,30,8,14,23,12,6,24,5,10)(3,15,17,28,20,26,32,31,21,29,16,9,19,25,18,27);(1,3,5,18,6,19,23,16,8,21,22,32,7,20,4,17)(2,9,12,26,13,27,10,29,14,28,11,25,24,31,30,15)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=f,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×#7162, #126701, #126722, … |
| maximal normal subgroup of | C2×#7162, #126701, #126722, … |
| maximal quotient of | #92919, #92920, #92921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 150);