Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #63091 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C24×D4 |
| commutator, Frattini, cosocle | C4 | C26 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #585 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
5349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2,7,12)(3,13,17,34)(4,9,19,30)(5,10,21,32)(6,23,25,11)(8,24,28,16)(14,35,38,55)(15,36,40,56)(18,29,42,53)(20,45,44,31)(22,48,47,33)(26,46,51,39)(27,49,52,41)(37,57,58,64)(43,61,60,54)(50,62,63,59);(1,3,7,17)(2,8,12,28)(4,14,19,38)(5,15,21,40)(6,24,25,16)(9,26,30,51)(10,27,32,52)(11,34,23,13)(18,37,42,58)(20,46,44,39)(22,49,47,41)(29,50,53,63)(31,55,45,35)(33,56,48,36)(43,62,60,59)(54,64,61,57);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,26)(10,29)(11,31)(12,30)(13,35)(15,37)(16,39)(17,38)(21,42)(22,43)(23,45)(24,46)(25,44)(27,50)(28,51)(32,53)(33,54)(34,55)(36,57)(40,58)(41,59)(47,60)(48,61)(49,62)(52,63)(56,64);(1,5)(2,10)(3,15)(4,18)(6,22)(7,21)(8,27)(9,29)(11,33)(12,32)(13,36)(14,37)(16,41)(17,40)(19,42)(20,43)(23,48)(24,49)(25,47)(26,50)(28,52)(30,53)(31,54)(34,56)(35,57)(38,58)(39,59)(44,60)(45,61)(46,62)(51,63)(55,64);(5,21)(10,32)(15,40)(18,42)(22,47)(27,52)(29,53)(33,48)(36,56)(37,58)(41,49)(43,60)(50,63)(54,61)(57,64)(59,62);(4,19)(9,30)(14,38)(18,42)(20,44)(26,51)(29,53)(31,45)(35,55)(37,58)(39,46)(43,60)(50,63)(54,61)(57,64)(59,62)Polycyclic presentation:
2^8|a^2=h,b^2=h,g^2=h,b^a=bg,g^a=gh,g^b=gh,f^c=fh,e^d=eh82 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56081);