Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #367 |
| 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:
(1,2,5,12,9,15,8,34)(3,16,18,48,21,50,27,49)(4,13,23,43,26,46,25,60)(6,29,28,45,24,11,31,42)(7,32,22,44,30,41,33,14)(10,35,36,17,40,52,39,54)(19,57,55,38,53,47,61,63)(20,59,56,37,58,62,51,64);(1,3,6,19,30,58,23,35,5,27,28,61,33,56,26,54,9,21,24,53,7,20,25,52,8,18,31,55,22,51,4,17)(2,10,13,37,44,63,42,49,34,36,60,62,32,57,11,16,15,40,46,64,14,38,45,48,12,39,43,59,41,47,29,50)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,d^2=h,e^2=fh,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fg,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94696, #94697, C2×#7476, … |
| maximal normal subgroup of | #94696, #94697, C2×#7476, … |
| maximal quotient of | #94024, #94046, #94205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 464);