Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D16 |
| commutator, Frattini, cosocle | C16 | C24 |
| socle | C2 | C22×D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14)(4,10)(5,23)(6,27)(7,9)(8,16)(11,25)(12,21)(13,22)(15,42)(17,47)(18,49)(19,41)(20,50)(24,39)(26,55)(28,37)(29,36)(30,46)(31,43)(32,57)(33,59)(34,54)(35,52)(38,63)(40,45)(44,61)(48,64)(51,60)(53,56)(58,62);(1,3)(2,8)(4,15)(5,24)(6,28)(7,19)(9,30)(10,31)(11,37)(12,39)(13,40)(14,23)(16,25)(17,21)(18,22)(20,51)(26,58)(27,32)(29,44)(33,61)(34,62)(35,47)(36,60)(38,64)(41,55)(42,50)(43,52)(45,63)(46,59)(48,54)(49,56)(53,57);(2,9)(8,30)(10,13)(11,33)(12,34)(14,41)(23,55)(27,36)(31,40)(32,60)(35,38)(37,61)(39,62)(42,49)(47,64)(50,56);(1,4,7,22)(2,10,9,13)(3,15,19,18)(5,20,26,53)(6,21,29,54)(8,31,30,40)(11,35,33,38)(12,36,34,27)(14,42,41,49)(16,43,46,45)(17,44,48,28)(23,50,55,56)(24,51,58,57)(25,52,59,63)(32,39,60,62)(37,47,61,64)Polycyclic presentation:
2^8|d^2=h,e^2=fg,f^2=gh,g^2=h,b^a=be,c^a=ch,e^a=ef,f^a=fg,g^a=gh,e^b=ef,f^b=fg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152533, #152536, #152539, … |
| maximal normal subgroup of | #152533, #152536, #152539, … |
| maximal quotient of | #512690, #512716, #512761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26969);