Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | Aut(G) = #72314138 (order 4096) · Inn(G) = #469 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2087 |
| prime radical | C23 | C2×Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,23,34,52,61,7,11,25,35,6,12,24,36)(3,8,14,28,42,57,62,64,15,30,43,59,16,29,44,58)(4,17,20,37,26,53,51,31,19,46,49,32,21,33,22,9)(13,38,39,60,45,63,50,54,18,47,48,55,40,56,41,27);(1,3)(2,8)(4,18,19,13)(5,14)(6,16)(7,15)(9,27,31,54)(10,28)(11,30)(12,29)(17,38,46,47)(20,48,49,39)(21,45,26,40)(22,50,51,41)(23,42)(24,44)(25,43)(32,55,37,60)(33,56,53,63)(34,57)(35,59)(36,58)(52,62)(61,64);(1,4,6,26,7,19,23,21)(2,9,12,37,11,31,34,32)(3,13,16,45,15,18,42,40)(5,22,24,20,25,51,52,49)(8,27,29,60,30,54,57,55)(10,33,36,17,35,53,61,46)(14,41,44,39,43,50,62,48)(28,56,58,38,59,63,64,47)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=dg,e^2=gh,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93818, #93819, #151506, … |
| maximal normal subgroup of | #93818, #93819, #151506, … |
| maximal quotient of | #146151, #146159, #146178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6508);