Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
542 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 542 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:
(1,2,8,14)(3,15,19,43)(4,20,24,35)(5,13,7,11)(6,26,29,38)(9,31,34,18)(10,21,37,23)(12,25,40,28)(16,44,48,62)(17,49,30,39)(22,53,56,36)(27,41,57,33)(32,54,60,64)(42,45,63,47)(46,50,58,59)(51,52,61,55);(1,3)(2,9)(4,16)(5,17)(6,27)(7,30)(8,19)(10,32)(11,33)(12,39)(13,41)(14,34)(15,26)(18,28)(20,50)(21,45)(22,54)(23,47)(24,48)(25,31)(29,57)(35,59)(36,62)(37,60)(38,43)(40,49)(42,51)(44,53)(46,55)(52,58)(56,64)(61,63);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,32)(11,35)(12,36)(13,20)(14,37)(15,42)(17,45)(18,46)(19,48)(25,52)(26,51)(27,54)(28,55)(29,56)(30,47)(31,58)(33,59)(34,60)(38,61)(39,62)(40,53)(41,50)(43,63)(44,49)(57,64);(1,5,8,7)(2,11,14,13)(3,17,19,30)(4,21,24,23)(6,25,29,28)(9,33,34,41)(10,35,37,20)(12,38,40,26)(15,39,43,49)(16,45,48,47)(18,27,31,57)(22,52,56,55)(32,59,60,50)(36,61,53,51)(42,62,63,44)(46,54,58,64)Polycyclic presentation:
2^8|a^2=h,d^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gh,e^b=eg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #495738, #498127, #499786, … |
| maximal normal subgroup of | #495738, #498127, #499786, … |
| maximal quotient of | #484216, #484218, #484220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26530);