Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #357 |
| commutator | C16 | C42 |
| Frattini, cosocle | C22×C16 | C22 |
| socle | C2 | #1380 |
| prime radical | C23 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,8,15,28,14)(3,16,18,48,21,39,53,49)(4,22,25,45,26,12,29,43)(6,10,27,40,30,42,61,41)(7,31,24,13,32,44,60,46)(9,33,34,23,38,59,37,17)(19,54,52,64,51,47,62,36)(20,55,50,35,56,63,58,57);(1,3,6,19,7,20,29,33,28,53,61,62,60,58,26,17,8,21,30,51,32,56,25,59,5,18,27,52,24,50,4,23)(2,9,12,35,13,36,41,48,11,34,43,63,44,54,10,39,15,38,22,57,46,64,40,49,14,37,45,55,31,47,42,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=f,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7479, #151968, #151973, … |
| maximal normal subgroup of | C2×#7479, #151968, #151973, … |
| maximal quotient of | #94025, #94049, #94050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 467);