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) = C2×C4 |
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 | D16 |
| 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 | |
85 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)(3,15,17,43)(4,11)(5,25)(6,28)(7,31)(8,14)(9,16,32,46)(10,22)(12,30)(13,29)(18,51,47,42)(19,53,49,64)(20,55,50,44)(21,38)(23,58)(24,41)(26,36)(27,37)(33,45,60,54)(34,62,61,52)(35,63,59,48)(39,57)(40,56);(1,3,5,18,26,49,56,63,29,50,57,62,30,54,24,46,8,17,22,47,6,19,23,48,7,20,27,52,21,45,4,16)(2,9,11,33,38,61,37,44,31,59,58,64,28,51,10,15,14,32,41,60,12,34,39,55,13,35,40,53,36,42,25,43)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,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 | #94694, #94695, C2×#7475, … |
| maximal normal subgroup of | #94694, #94695, C2×#7475, … |
| maximal quotient of | #94024, #94047, #94203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 463);