Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13691 |
| automorphism | Aut(G) order 4096 |
| 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 | #508 | C22 |
| 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 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,9,16)(3,14,20,35)(4,21,24,53)(5,15,8,12)(6,26,30,58)(7,10,19,36)(11,37,39,23)(13,41,44,28)(17,47,48,64)(18,40,50,43)(22,54,32,52)(25,34,29,57)(27,45,59,62)(31,42,51,63)(33,60,61,56)(38,55,46,49);(1,3,8,19,9,20,5,7)(2,10,15,35,16,36,12,14)(4,17,32,51,24,48,22,31)(6,18,29,41,30,50,25,28)(11,33,46,62,39,61,38,45)(13,34,43,58,44,57,40,26)(21,42,54,64,53,63,52,47)(23,27,49,60,37,59,55,56);(1,4)(2,11)(3,17)(5,22)(6,27)(7,31)(8,32)(9,24)(10,33)(12,38)(13,42)(14,45)(15,46)(16,39)(18,49)(19,51)(20,48)(21,26)(23,28)(25,56)(29,60)(30,59)(34,54)(35,62)(36,61)(37,41)(40,47)(43,64)(44,63)(50,55)(52,57)(53,58)Polycyclic presentation:
2^8|a^2=h,b^2=dh,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 | #151526, #151527, #151529, … |
| maximal normal subgroup of | #151526, #151527, #151529, … |
| maximal quotient of | #151624, #151625, #151630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6681);