Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2408 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
635 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 635 conjugacy classes of subgroups — too many to draw. Only its 133 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)(3,14)(4,19)(5,25)(6,29)(7,12)(8,13)(9,17)(10,23)(11,26)(15,31)(16,44)(18,33)(20,35)(21,36)(22,38)(24,37)(27,39)(28,53)(30,34)(32,46)(40,45)(41,59)(42,56)(43,57)(47,49)(48,63)(50,52)(51,60)(54,58)(55,61)(62,64);(1,3,6,17,8,18,7,30)(2,9,12,14,13,34,29,33)(4,20,22,15,24,42,23,43)(5,16,26,45,28,46,27,54)(10,35,37,57,38,56,19,31)(11,32,25,40,39,44,53,58)(21,48,49,41,51,61,50,62)(36,59,52,63,60,64,47,55);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,31)(11,36)(12,37)(13,38)(14,35)(16,41)(17,42)(18,43)(19,29)(20,30)(25,52)(26,49)(27,50)(28,51)(32,55)(33,56)(34,57)(39,60)(40,59)(44,63)(45,61)(46,62)(47,53)(48,54)(58,64);(1,5,7,27,8,28,6,26)(2,11,29,53,13,39,12,25)(3,16,30,54,18,46,17,45)(4,21,23,50,24,51,22,49)(9,32,33,58,34,44,14,40)(10,36,19,47,38,60,37,52)(15,41,20,48,43,62,42,61)(31,55,56,64,57,63,35,59)Polycyclic presentation:
2^8|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,f^a=fh,c^b=cg,g^b=gh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #499778, #499780, #499783, … |
| maximal normal subgroup of | #499778, #499780, #499783, … |
| maximal quotient of | #263036, #263040, #263043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17284);