Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13691 |
| automorphism | Aut(G) = C2×#72216771 (order 2048) · Inn(G) = #2206 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2206 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | #2263 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
100 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)(3,13)(4,19)(5,24)(6,27)(7,31)(8,14)(9,16)(10,21)(11,26)(12,29)(15,44)(17,48)(18,43)(20,53)(22,52)(23,54)(25,34)(28,42)(30,60)(32,40)(33,57)(35,55)(36,61)(37,49)(38,56)(39,50)(41,47)(45,51)(46,64)(58,59)(62,63);(1,3,5,16,8,18,26,7)(2,9,11,13,14,31,24,43)(4,15,20,45,23,46,22,32)(6,17,25,47,30,50,58,29)(10,33,35,42,38,36,37,63)(12,34,39,27,41,59,48,60)(19,51,52,44,54,40,53,64)(21,28,49,57,56,62,55,61);(1,4)(2,10)(3,15)(5,20)(6,28)(7,32)(8,23)(9,33)(11,35)(12,40)(13,42)(14,38)(16,45)(17,49)(18,46)(19,27)(21,29)(22,26)(24,37)(25,57)(30,62)(31,36)(34,53)(39,64)(41,51)(43,63)(44,48)(47,56)(50,55)(52,59)(54,60)(58,61)Polycyclic presentation:
2^8|b^2=d,d^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13691, #512928, #512929, … |
| maximal normal subgroup of | C2×#13691, #512928, #512929, … |
| maximal quotient of | #151618, #151619, #151622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6679);