Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2416 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 405 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,29,8,13,7,12)(3,14,17,9,19,33,18,34)(4,20,30,10,24,37,23,38)(5,25,26,11,28,53,27,39)(15,31,21,35,43,57,42,56)(16,44,45,40,47,32,46,58)(22,36,54,52,51,60,50,48)(41,59,49,63,62,64,61,55);(1,3,6,17,8,19,7,18)(2,9,12,14,13,34,29,33)(4,21,23,15,24,42,30,43)(5,16,26,45,28,47,27,46)(10,35,37,57,38,56,20,31)(11,32,25,40,39,44,53,58)(22,49,50,41,51,61,54,62)(36,59,52,63,60,64,48,55);(1,4)(2,10)(3,15)(5,22)(6,23)(7,30)(8,24)(9,31)(11,36)(12,37)(13,38)(14,35)(16,41)(17,42)(18,21)(19,43)(20,29)(25,52)(26,50)(27,54)(28,51)(32,55)(33,56)(34,57)(39,60)(40,59)(44,63)(45,61)(46,49)(47,62)(48,53)(58,64);(1,5,7,27,8,28,6,26)(2,11,29,53,13,39,12,25)(3,16,18,46,19,47,17,45)(4,22,30,54,24,51,23,50)(9,32,33,58,34,44,14,40)(10,36,20,48,38,60,37,52)(15,41,21,49,43,62,42,61)(31,55,56,64,57,63,35,59)Polycyclic presentation:
2^8|a^2=fh,b^2=e,d^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501254, #501255, #501271, … |
| maximal normal subgroup of | #501254, #501255, #501271, … |
| maximal quotient of | #277853, #277855, #277859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17798);