Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #367 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | C2×#367 |
| prime radical | C23 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
91 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)(3,14,18,41)(4,9)(5,23)(6,27)(7,13)(8,17,31,51)(10,25)(11,21)(12,22)(15,34)(16,39,48,43)(19,54,53,61)(20,55)(24,38)(26,57)(28,35)(29,56)(30,47,33,60)(32,44,59,64)(36,46)(37,45)(40,62,58,52)(42,50,63,49);(1,3,15,44,6,19,5,17,22,48,25,52,24,50,56,60,7,18,46,64,28,53,26,51,4,16,45,62,21,49,20,47)(2,8,29,54,11,32,10,14,9,30,23,42,35,40,34,39,13,31,55,61,38,59,37,41,12,33,57,63,27,58,36,43);(1,4,7,22)(2,9,13,12)(3,16,18,48)(5,20,26,56)(6,21,28,24)(8,30,31,33)(10,34,37,36)(11,35,38,27)(14,39,41,43)(15,45,46,25)(17,47,51,60)(19,49,53,50)(23,55,57,29)(32,40,59,58)(42,61,63,54)(44,62,64,52)Polycyclic presentation:
2^8|a^2=e,b^2=dfg,c^2=h,d^2=fgh,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #151998, #151999, #152300, … |
| maximal normal subgroup of | #151998, #151999, #152300, … |
| maximal quotient of | #151037, #151038, #151054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6626);