Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30871 |
| automorphism | Aut(G) = #72340866 (order 12288) · Inn(G) = #2675 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2675 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2017 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
413 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)(4,20)(5,11)(6,30)(7,12)(8,13)(9,17)(10,24)(15,44)(16,47)(18,34)(19,33)(21,43)(22,53)(23,38)(25,37)(26,58)(27,41)(28,40)(29,42)(31,46)(32,48)(35,50)(36,55)(39,49)(45,63)(51,56)(52,62)(54,59)(57,61)(60,64);(1,3)(2,9)(4,21)(5,26)(6,18)(7,17)(8,19)(10,35)(11,39)(12,33)(13,34)(14,30)(15,23)(16,27)(20,31)(22,54)(24,46)(25,50)(28,49)(29,48)(32,40)(36,59)(37,44)(38,43)(41,58)(42,47)(45,56)(51,62)(52,55)(53,60)(57,64)(61,63);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,37)(13,38)(14,43)(16,45)(17,21)(18,50)(19,46)(20,30)(26,52)(27,55)(28,57)(29,56)(32,59)(33,35)(34,44)(39,62)(40,53)(41,51)(42,61)(47,60)(48,64)(49,54)(58,63);(1,5,6,27,8,29,7,28)(2,11,12,40,13,42,30,41)(3,16,17,26,19,49,18,48)(4,22,23,55,25,56,24,57)(9,32,14,39,34,58,33,47)(10,36,37,53,38,61,20,51)(15,45,21,52,46,54,50,64)(31,59,43,62,44,63,35,60)Polycyclic presentation:
2^8|d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #30871⋊C3 |
| maximal subgroup of | #151265, #151267, #151269, … |
| maximal normal subgroup of | #151265, #151267, #151269, … |
| maximal quotient of | #339703, #339706, #339712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23859);